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PUBLISHERS^  NOTICE. 


FRENCH'S   ARITHMETICS. 

This  Series  consists  of  Five  Books,  viz, : 

I.    FIRST   LESSONS   IN   NUMBERS. 

II.   ELEMENTARY   ARITHMETIC, 
in.   MENTAL  ARITHMETIC. 
IV.   COMMON   SCHOOL  ARITHMETIC. 

V.   ACADEMIC   ARITHMETIC.     (lu  Preparation.) 


Tlie  Publishers  present  this  Series  of  Text-Books  to  American  Teach- 
ers, fully  believing  that  they  contain  many  new  and  valuable  features  that 
will  especially  commend  them  to  the  practical  wants  of  the  age. 

The  plan  for  the  Series,  and  for  each  book  embraced  in  it,  was  fully 
matured  before  any  one  of  the  Series  was  completed ;  and  as  it  is  based 
upon  true  philosophical  principles,  there  is  a  harmony,  a  fitness,  and  a 
real  progressiveness  in  the  books,  that  are  not  found  in  any  other  Series 
of  Arithmetics  published. 


Entered,  according  to  Act  of  Congress,  in  the  year  1869,  by 

HARPER    &    BROTHERS, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the 

Southern  District  of  New  York. 


■'^-^^^^^ 


^ ==:^ 


Business  men  generally  agree  in  the  statement  that,  after  leaving  school. 
they  were  obliged  to  learn,  and  in  many  cases  to  devise  for  themselves, 
methods  of  computation  adapted  to  their  use  in  business.  The  universality 
of  this  experience  led  the  Author  of  this  Series  of  Arithmetics  to  a  careful 
and  protracted  investigation  into  the  philosophy  of  the  development  of 
the  mathematical  powers  of  the  mind,  and  to  a  critical  examination  into 
the  present  methods  of  teaching  Arithmetic.  From  these  investigations 
he  has  become  convinced  that,  in  order  to  make  practical  arithmeticians^ 
a  radical  change  in  the  plan  of  text-boolis  upon  the  subject  is  necessary. 

The  acknowledged  requisites  of  a  good  method  of  instruction  are  these: 

1st.  It  must  be  adapted  to  the  nature  of  both  subject  and  learner ; 

2i.  It  must  be  an  uninterrupted  progress  from  the  easy  to  the  difficult ; 

3i.  It  must  use  facts  known  to  the  learner,  in  imparting  to  him  a  knowl- 
edge of  the  unknown  subject ; 

4ih.  It  must  regard  the  natural  order  of  development  of  the  subject,  and 
present  it  in  that  order ; 

hth.  It  must  arrange  the  substance  of  the  facts  presented  under  each 
general  division,  into  brief  summaries,  recapitulations,  or  general  principles ; 

Qth.  It  must  thoroughly  reach  the  understanding  of  the  learner. 

This  book  fully  recognizes  these  requisites,  in  every  Chapter,  Section, 
and  Case.  The  attention  of  teachers  and  parents  is  particularly  invited  to 
the  following  distinctive  features  of  the  work : 

Order  of  Subjects. — A  philosophical  arrangement  of  subjects  has  been 
carefully  observed.  The  four  classes  of  numbers,  Integers,  Decimals,  Com- 
pound Numbers,  and  Fractions,  are  presented  in  the  order  here  stated. 
But  Factors  and  Multiples  precede  Fractions,  as  a  knowledge  of  the  former 
subject  is  essential  to  a  clear  understanding  of  the  latter.  The  Chapter 
'1  Converse  Operations  presents,  in  immediate  connection,  those  operations 
J?,  numbers  which  are  the  converse  of  each  other.  The  succeeding  Chapters 
>i'esent  the  subjects  of  Percentage,  Ratio  and  Proportion,  Evolution,  Pro- 
gressions, Mensuration,  and  Miscellaneous  Problems. 

Principles  and  Rules. — Each  new  process  or  method  of  computation 
is  introduced  inductively,  and  the  Principles  which  follow  are  evident 
sequents  of  the  Inductions.  The  learner  is  required  to  make  practical 
^applications  of  these  Principles  in  solving  a  number  of  Problems,  thus 
giving  him  a  thorough  comprehension  of  the  principles  upon  which  the 
practical  calculations  of  after  life  are  based.  Rules  are  then  presented, 
v.'hich  arc  uniformly  based  upon  the  Principles  previously  established. 


IV  PKEFACE. 

IllustTations.— The  pictures  are  designed  to  aid  the  pupil  in  acquir- 
ing a  clear  understanding  of  the  subjects  and  principles  which  they  illustrate, 
and  also  to  cultivate  a  taste  for  the  beautiful  in  art.  The  Italic  Figures, 
which  were  cut  expressly  for  this  Scries,  also  add  much  to  the  beauty  and 
attractiveness  of  the  book. 

Problems.— Much  labor  has  been  bestowed  upon  the  preparation  of  the 
Problems,  to  make  them  the  vehicle  of  practical  knowledge.  Nearly  every 
occupation,  trade,  profession,  and  art  has  its  own  peculiar  business  terms, 
and  its  peculiar  articles  of  commercial  exchange,  with  their  appropriate 
market  values.  The  use  of  these  business  terms,  and  transactions  in  these 
articles  of  exchange,  make  up  the  every-day  matters  of  the  business  w^orld. 
Thus,  the  merchant,  the  manufacturer,  the  grocer,  the  druggist,  the  phy- 
sician, the  lawyer,  the  printer,  the  bookseller,  etc.,  etc.,  has  each  his  own 
statistics  of  business  ;  and  from  these  have  been  prepared  problems  that 
convey  to  the  pujsil  a  great  amount  of  knowledge  of  the  principles,  customs, 
and  details  of  business.  Such  problems  are  all  the  more  interesting  from 
their  practical  utility,  while  they  are  none  the  less  illustrative  of  the  Prin- 
ciples of  Numbers.  The  Problems  are  all  prepared  from  materials  collected 
for  this  book,  and  present  statements  and  business  terms  in  a  correct 
business  way.  In  this  way  the  arithmetic  of  the  school-room  may  be  made 
to  meet,  in  a  considerable  degree,,  the  practical  wants  of  after  life.  Abstract 
problems  possess  little  interest  for  pupils,  and  hence  few  of  them  are  found 
in  this  book  :  but  the  number  of  practical  problems  drawn  from  the  every- 
day transactions  of  business,  greatly  exceeds  that  in  any  similar  work. 

Useless  Matter. — Reduction  of  State  Currencies  is  nearly  obsolete, 
Alligation  Alternate  is  merely  a  curiosity  of  numbers,  English  Money  is  of 
little  value  to  American  youth,  and  many  denominations  in  Compound 
Numbers  often  heard  of  in  the  school-room  are  unknown  in  business. 
Those  who  look  for  these  matters  in  this  book  will  look  in  vain. 

New  and  Distinctive  Features  will  be  found  in  the  Notation, 
Multiplication,  and  Division  of  Integers  and  Decimals  ;  Compound  Num- 
bers ;  Factors  and  Multiples ;  Division  of  Fractions ;  Converse  Operations ; 
Table  of  Legal  Eates  of  Interest,  from  official  sources;  Classification  of  all 
computations  in  Percentage  under  Five  Ceneral  Cases  ;  Rules  for  Interest 
and  Partial  Payments ;  Average  of  Accounts ;  Proportion;  Evolution;  Pro- 
gressions; Review  and  Miscellaneous  Problems;  and  the  deductions  of 
Principles  from  Inductions,  and  the  basing  of  Rules  upon  Principles. 

In  the  preparation  of  this  work  for  American  Schools,  the  Author  has 
had  constantly  in  mind  the  present  condition  and  the  "future  requirements 
of  American  youth.  The  work  differs,  both  in  general  plan  and  details, 
from  other  works  upon  the  same  subject.  It  is  confidently  believed  that 
the  adoption  and  introduction  into  schools  of  the  distinctive  features  of  the 
book  will  eliect  a  change  in 
making  good^  pra-^Aical  AritJiimticiam, 


A  "Word  with.  Teacliers, — A  hint,  a  snggestion^^r  an  item  of  information  not 
found  in  the  body  of  a  text-book,  -will  often  awaken  thonglit,  and  start  a  train  of  inqui- 
ries in  a  class,  that  will  greatly  increase  their  interest  in  the  study.  Connected  with  the 
subject  Arithmetic,  and  not  found  in  text-books,  are  many  items  of  interest  and  im- 
portance to  pupils,  to  which  their  attention  should,  at  the  proper  time,  be  directed.  This 
Manual  is  intended  to  give  you  brief  hints  and  suggestions,  that  will  enable  you  the 
better  to  give  instruction  to  your  pupils  in  this  important  branch  of  study. 

Page  9.  Arithmetic,  as  a  Science,  logically  investigates  and  philosophically  classifies 
and  arranges  the  principles  and  rules  of  the  subject ;  as  an  Art,  it  applies  the  principles 
and  rules  for  computations,  to  the  practical  affairs  of  lire. 

10.  As  the  Roman  Notation  is  not  presented  in  this  book,  it  may  be  well  to  spend 
time  enough  in  giving  oral  instruction  upon  the  subject,  to  make  pupils  familiar  with 
the  following  facts  and  their  application : 

I.  The  Roman  Notation  employs  seven  letters  to  represent  numbers. 

11.  Each  letter  has  a  fixed  value  when  used  alone.  Thus,  I. =1,  V.  =5,  X.  =10,  L.  =50, 
C.=100,  D.=500,  and  M.  =  1,000. 

III.  Repeating  a  letter  repeats  its  value.    Thus,  II.=2,  XXX.=30,  CC.=200. 

IV.  Whejb  any  letter  stands  after  another  expressing  greater  value,  the  number  ex- 
pressed is  the  sum  of  the  values  of  the  two  letters.    Thus,  VI.=6,  XV,=15,  CXVI.=116. 

V.  When  any  letter  stands  before  another  expressing  greater  value,  the  number  ex- 
pressed is  the  difference  of  the  values  of  the  ttco  letters.  Thus,  IV. =4,  IX. =9,  XC.=90, 
CM.  =900. 

VI.  When  a  letter  stands  betioeeniwn  others,  loth  of  greater  value,  its  value  is  taken 
from  the  sum  of  their  values.    Thus,  XIX.=19,  CIV.=104,  MXL.  =1,010. 

13.  Beginners  require  abundant  practice  both  in  writing  and  reading  numbers.  Give 
them  numerous  exercises  on  each  new  period  of  figures. 

15.  Explain  clearly  that  the  simple  value  is  the  number  of  ones  or  units  expressed  by 
the  figure,  and  the  local  value  is  the  value  given  to  these  units  by  the  place;  i.  e.,  one 
is  the  value  of  the  figure  determined  by  its  form,  the  other  by  the  place  it  occupies. 

Bull  pupils  may  be  aided  in  learning  to  write  and  read  numbers,  by  allowing 
them  at  first  to  write  a  Skeleton  of  JVotaiion,  consisting  of  periods  of  ciphers;  thus, 
000,000,000,000;  and  under  this  to  write  the  figures  of  any  given  number,  in  their 
proper  places.  They  should  also  learn  the  Family  Name  of  each  period, — as  ones,  thou- 
sands, millions;  and  the  nam.es  of  ike  places  in  each  family, — as  ones,  tens,  hundreds. 

19,  Give  original  problems  under  each  Section  and  Case  in  this  Chapter,  before  as- 
signing the  problems  from  the  book.  This  may  be  done  in  various  ways  ;— the  following 
has  been  found  a  good  one :  At  the  close  of  a  recitation,  or  other  convenient  time,  put 
problems  on  the  blackboard,  and  let  the  class  copy  them  upon  their  slates.  At  the  com- 
mencement of  the  next  recitation,  call  for  the  results  (and  solutions  also,  if  you  choose), 
to  all  the  problems  given  out  at  the  previous  lesson, 

25.  More  mistakes  occur  in  addition  than  in  any  other  process.  Thorough  drills  in 
adding  columns  whose  sums  reach  100,  will  greatly  lessen  this  defect.  These  26  problems 
(41—67)  afford  this  kind  of  drill ;  they  sliould  not  be  skipped. 


VI  ,  MANUAL. 

30.  Those  teachers  who  prefer  to  use  the  "borrowing  10"  method  of  subtraction, 
will  find  an  explanation  of  that  method  in  the  Elementary  Arithmetic  of  this  Series. 

32.  An  explanation  oi  Left-hand  Subtraction  sometimes  assists  pupils  in  acquiring  a 
clear  understanding  of  the  process  commonly  called  "  borrowing  ten."  The  following 
course  will  enable  you  to  make  the  explanation : 

\st.  Solve  the  problems  in  Case  I.,  page  28,  commencing  at  the  left  hand  to  subtract 

2(7.  Take  a  subtrahend  whoso  right-hand  figure  only  exceeds  the  corresponding  figure 
i.i  the  minuend  (as  582 — 347),  commence  at  the  left  to  subtract,  calling  the  tens  figure  of 
the  minuend  1  less,  and  adding  the  1,  as  a  ten,  to  the  ones  of  the  minuend. 

M.  Proceed  in  a  simila»manner  with  problems  in  which  other  orders  of  units  of  the 
subtrahend  exceed  the  like  orders  in  the  minuend,  as  shown  in  the  following; 

Explanation.— Since  7  is  more  than  3,  Ave  have  4—1  =  3.  Since  2  is  more  solution. 
than  0,  we  have  12—7—5.  Since  4  is  not  more  than  6,  we  have  10—2=8.  53062 
Since  8  is  more  than  2,  we  have  5— 4=1.     And  12— 8=4.  17248 

With  practice,  pupils  will  acquire  great  facility  in  left-hand  subtraction.       35814= 

34.  The  market  grades  or  qualities  of  some  kinds  of  goods  are  indicated  by  certain 
marks  upon  the  cask  or  package.  Thus,  in  sugars,  we  have  A  or  "  straight  A,"  4ior 
'•  diamond  A,"  ®  or  "  circle  A,"  B,  and  C ;  etc.  V 

38.  In  explanations  of  solutions,  use  the  true  multiplicand  for  the  multiplicand. 

52.  A  clear  understanding  of  this  Principle  will  aid  the  pupil  in  the  induction  to 
division  of  decimals. 

57.  These  problems  should  be  solved  by  both  long  and  short  division. 

59.  In  long  division,  require  pupils  to  memorize  the  catch-words,  divide,  multiplij, 
subtract,  bring  down.     And  indelibly  impress  upon  their  memories  this  fact ; 

For  every  figure  of  the  dividend  used  after  the  first  quotient  figure  is  obtained, 
there  must  be  a  figure  in  the  quotient. 

61.  Referring  to  93,  Note  1,  teach  the  pupil  how  to  write  a  quotient  containing  a 
fraction;  as  142|. 

69.  Instruct  pupils,  when  explaining  solutions,  to  tell  what  is  given,  and  what  is  re- 
quired. Thus,  "  In  problem  4  are  given  a  number  and  all  its  2>arts  but  one.  To  find 
this  part,  I  subtract  the  sum  of  the  given  parts  from,  the  number."' 

In  problem'9  exjjlain  the  use  of  the  '*  or  repeating  marks. 

70.  Problem   16.    "  As  per   annexed  schedule "  is  a  {30). 
common  business  reference  to  a  bill  or  memorandum  on  Bountij,  $949 
the  same  paper.     Call  the  attention  of  pupils  to  the  signi-  ^  ''^°-  ®'  ^i^Z.  ^Vii 
fication  of  the  commercial  terms  found  in  the  problems.  -tn    n     n       yyZ      2'*! 

71.  -P^'oWem  30.  Cultivate  neatness,  order,  and  method  7  "  "  18—  126 
in  blackboard  work.  Thus,  the  solution  of  this  problem  36  mo.  $1512^ 
might  be  placed  upon  the  blackboard  as  here  shown.  $1512-*-36=$42 perm,o. 

Explain  the  meaning  of  Average,  and  how  it  is  found. 

72.  The  Areas  in  this  Table  are  from  standard  authorities.  A  little  ingenuity  will 
enable  you  to  greatly  increase  the  number  of  problems  of  the  kind  found  on  this  page. 

76.  The  only  place  in  which  and  is  properly  used  in  reading  numbers,  is  in  a  mixed 
mimber,  after  the  integer.     Thus,  5.7,  5  and  7  tenths;  '^J'-,  4  and  9  sixteenths. 
78.  Pupils  should  carefully  compare  Art.  129,  130,  with  Art.  25,  26,  27. 

80.  Familiarize  pupils  with  the  reading  of  such  numbers  as  1400.010  ;  1000.410  ;  .1400. 

81.  )  Instruct  pupils  to  place  the  decimal  point  in  the  result,  before  adding    57 
83.  )  or  subtracting  the  ones.     Require  them  to  subtract  without  writing       5.91 

decimal  ciphers  in  the  minuend  over  decimal  figures  in  the  subtrahend, 

94.  Encourage  pupils  to  solve  problems  in  different  ways.  Thus,  "  In  how  many 
ways  can  this  problem  (23)  be  solved ?"    "  Which  way  do  you  prefer,  and  why?" 


MANUAL.  Vii 

97.  First  Reference. — The  coins  are  shown  in  the  cut,  page  95,  in  perspective,  and  of 
course  only  the  longest  diameter  is  correct  in  measurement. 

Second  Reference.— Gold  and  silver  coins  are  alloyed,  to  make  them  hard  enough  for 
nse  as  a  circulating  medium,  without  depreciating  perceptibly  in  value  by  wear. 

Third  liefet^ence.—FuTpils  must  make  all  their  computations  in  decimals ;  and  express 
parts  of  a  cent,  in  final  results,  in  fractions,  when  they  are  halves,  fourths,  or  eighths. 

113.  "  Fixed  Standards"  are  weights  and  measures  established  by  the  General  Gov- 
ernment, or  recognized  and  sanctioned  by  custom. 

115.  Abbreviations  of  denominations  should  always  be  written  after  the  numbers, 
and  be  followed  by  periods.     The  signs  $  and  £  are  written  before  the  nnmbers. 

118.  By  English  Statute  Law,  a  heaped  bushel  is  18^- inches  in  diameter,  8  inches 
deep,  and  heaped  in  a  true  cone  to  the  height  of  6  inches.  Tliis  cone,  18|  inches  in 
diameter  and  6  inches  high,  is  1  peck. 

125.  )       The  denominations  of  square  and  cubic  measures  are  used  only  in  computa- 

126.  )  tions,  the  measurements  being  taken  in  linear  units. 

131.  Each  lot  of  a  Government  Section  of  land  is  divided  into  2  Forties.     Hence, 
1  section  (640  A.)=4  quarter-sections  (160  A.  each)=8  80-acre  lots  (or  80's)=16  40's. 
135.  1  solar  year    is  5  h.  4S  min.  48  sec.  longer  than  a  common  year. 

4      "     years  are  23  "  15    "      12    "        "  4        "        years. 

100      "        "       "      24  da.  5  «  20    "  "  100        "  " 

400      "        "       "      96  "  21   "  20    "  "  400        "  " 

Hence,  if  97  days  be  added  to  every  400  years,  the  calendar  will  be  only  2  h.  40  min. 
ahead  of  true  time.     These  9T  days  are  distributed  among  97  leap-years.    (See  234.) 

137.  Require  pupils  to  point  out,  on  all  diagrams,  the  lines  defined. 

138.  A  geogi-aphic  mile=Jjj  of  69.16  Eng.  mi.=l'.15T*j  mi.=l  mL  48.85  rd. 

142.  The  commonly  recognized  units  of  the  other  Tables  are,  for  Canada  Money, 
Dollar;  Sterling  Money,  Pound;  Wood  Measure,  Cord;  Surveyors'  Linear  Measure, 
Foot  and  Chain ;  Surveyors*  Square  Measure,  Square  Chain  and  Acre ;  Time,  Day  and 
Year ;  Circles,  Degree,  Right  Angle,  and  Circumference. 

144.  As  none  of  the  tables  or  denominations  of  the  Metric  System  have  come  into 
actual  use,  a  presentation  of  theTables  is  all  that  the  present  state  of  the  subject  demands. 

150.  )  Problems  like  17  are  easily  solved  by  left-hand  subtraction. 

152.  >  Commencing  at  the  left,  we  have  17  mi. — 14  mi, 
=3  mi.   3  rd.  (4  rd.-l  rd.)— 0  rd.=3  rd.    5i  yd.  +  2  yd.    ^7  mi.  4  rd.  2  yd.  1  ft. 
r-7iyd.,and7iyd.— 4yd.=3iyd.=3yd.lft.6in.    1  ft,        ^         *^        ^        ^ 


6  in.  + 1  ft.  =  2  ft.  6  in.,  and  2  ft  0  in.-2  ft. =0  ft.  6  in.  ^  "**•  3rd.3yd.O  ft.  6  in. 

167.  If  the  multiplier  or  divisor  is  less  than  1,  the  first  two  principles  will  be  re- 
versed. 

178.  The  following  method  of  finding  the  least  common  multiple  is  preferred  by 
many.     Take,  for  example,  Problem  13,  page  178. 

We  first  arrange  the  numbers  from  least  to  great-    J?,  i>^,  'i^,  J^,  35,  45,  60,  72  5 


est,  and  cancel  or  drop  such  as  are  factors  of  any  7,    9,  12,  72 


of  the  others.    We  next  divide  through  by  any  prime  7,    3,    4,  24 

number  that  is  a  factor  of  any  two,  or  by  any  number  Y,    Y,    4,     8 


that  is  a  factor  of  all  the  given  numbers;  and  divide  ^r     ^      j^     ^j 

these  results  and  the  undivided  numbers  in  the  same    >yy^2'>i.5')(.3y.3y.4:=2520 

manner ;  and  so  on,  until  the  final  quotients  are  prime  to 

one  another.  The  divisors  and  final  quotients  are  the  factors  of  the  least  common  multiple. 
183.  Show  wherein  these  General  Principles  are  similar  to  the  General  Principles  of 

Division,  Art.  278. 

187.  The  fractional  unit  J  is  6  times  as  great  as  the  fractional  unit  ^V    That  is, 
The  less  the  denominator  of  a  fraction,  Vie  greater  is  its  fractional  unit. 


VUl  MANUAL. 

193  Mixed  numbers  are  readily  subtracted  by  left-hand  subtraction.  7^  =  7s*s- 
For  example,  take  Ex.  2,  page  193.  Since  l^  is  more  than  ^^,  we  have  5|=3|2 
0-3=3,  and  -^r-tS^BS-  S^i 

206.  Give  several  original  problems  like  Problems  9,  10.  Require  pupils  to  write  out 
a  full  explanation  of  a  solution,  and  therefrom  deduce  the  Principle  stated  in  this  Note. 

211.  Decimal  figures  which  continually  repeat,  are  called  a  Repetend.  Its  value  is 
expressed  by  a  fraction  with  the  repeating  figures  for  the  numerator,  and  as  many  9's 
for  a  denominator.     Thus,  i  =  .66G..  ..  =  |  =  ?  ;  |=.428571. . .  .  =  ||fi|i  =  ?. 

221.  C.  is  the  abbreviation  for  the  Latin  Centum,  signifying  one  hundred;  and  M. 
for  the  French  Mille,  signifying  one  thousand. 

223.  A  logical  explanation  of  any  solution  requiring  more  than  one  process  or  com- 
putation, or  of  the  reasons  upon  which  any  principle  or  process  is  based,  is  an  Analysis. 
This  section  applies  particularly  to  the  solution  of  problems  which  involve  more  than 
one  process  or  computation. 

2.34.  Pupils  should  face  south,  and  hold  their  books  erect  before  thera,  while  study- 
ing or  explaining  this  astronomical  cut. 

252.  The  tax  on  any  sum  from  $1  to  $10,000  can  be  taken  from  a  Table  that  gives 

only  the  tax  on  $1,  $2,  $3.. to $10  inclusive,  if  the  table  is  carried  to  six  decimal 

plices.    Thus,  if  the  tax  on  $1  is  $.023145,  on  $10  it  is  $.23145;  on  $100,  $2.3145;  on 
$10  10,  $23,145;  on  $10,000,  $231.45;  and  so  of  $2,  $20,  $200,  $2,000,  etc. 

262.  Teachers  in  Vt.,  N.  II.,  or  Conn,  should  require  pupils  to  solve  the  problems  on 
page  262,  both  by  the  U.  S.  Court  Rule,  and  the  Rule  for  their  own  State. 

291.  Explain  that  the  third  term  is  divided  by  the  first,  to  find  the  value  of  a  unit  ; 
and  the  result  is  multiplied  by  the  second  term,  to  find  the  value  of  the  number  of 
units ;  the  same  as  in  Analysis,  Art,  370-  Also,  require  pupils  to  solve  the  problems 
by  analysis,  after  solving  them  by  proportion. 

293.  Pupils  will  learn  to  state  problems  very  rapidly,  if  they  are  taught  to  first  write 
the  terms  in  two  lines,  as  they  occur,  writing  the  second  set  of  conditions  under  cor- 
responding terms  of  the  first.     For  example,  in  Problem  G,  page  294,  the  pupil  writes 

10  h.        1365  bar.        13  da.  tt    ,       ., 

-.Q-u  ?      "  sa   '»  ^^^  ^^^^  ^^^^  ''"^y  *°  arrange  the  couplets. 

(P.  202,  prob.  24).  (P.  292,  prob.  24). 

In    solving   problems,   some  I  jSfi.    fi  2     ^/^-i   *   -     x  y 

teachers  write  the  work  in  one       '^  ^P  /  J's  J^^'     17  17  ^Sf     X-^ ^'^  '^ 

of  the  forms  here   shown,  in  ~^i  SXo  34lr^ 

preference  to  the  forms  given  68~ft  — 68^ fi 

on  pages  290,  293.  "^  '  "^  " 

300.  Other  roots  are  indicated  by  placing  over  the  radical  sign  the  figure  denoting 
the  required  root;  as  -\/}  -\/,  ^,  etc. 

310.  Since  2^  x  2»  =  2*,  the  V  of  a  number  =  V  of  V-  And  since  2'  X  2>  y,  2»  = 
(2»)»  =2»,  and  2»  X  2»  =  (2^)=  =  2«,  the  V  of  a  number  =  V  of  V  of  V?  or  V  of  V> 
or  V  of  V* 

313.  j    In  Progressions,  if  any  three  of  the  five  things  are  given,  the  other  two  may 

317.  )  be  found.   The  rules  here  given  cover  the  ordinary  applications  of  the  subject. 

321.  Have  a  figure  drawn  to  illustrate  each  definition  and  problem  in  this  Chapter. 

325.  i     These  Principles  (V.,  p.  325,  III.,  p.  327)  may  be  made  Dlain  to  the  pupil  by 

32  7.  J  practical  appU  cations. 


\ 


y    h  A  Uhii  is  a  single  thing,  or  one,  of  any  kind. 
y   2.  A  JVmnber  is  a  unit,  or  a  collection  of  units. 

Note. — Any  number  is  cither  concrete  or  abstract. 

V  3.  A  Conc7*ei:e  JV*umber  is  a  number  applied  to  some 
object ;  as,  four  men,  ten  apx^les,  fifty  days. 

y    4t  An  Abstract  JVumber  is  a  number  not  applied  to 
any  object  ;  as,  four,  ten,  fifty. 

V  5.  An  Jnteger  is   a  number  tbo  units  of  wbicb  arc 
whole  or  undivided. 

Note. — Integers  arc  also  called  ^ylLole  Numbers. 

y  6,  TTniiy  is  the  abstract  unit  1. 

7.  A.rWl7neHe  is  the  science  of  numbers,  and  the  art  of 

computation.     (See  Manual,  page  5.) 

8.  A  SotuHo7l  is  a  process  of  computation  used  to  ob- 
tain a  required  result. 

9.  A  "I^roblein  is  a  question  requiring  a  solution. 

'  10.  An  ^xptanaiioJi  is  a  statement  of  the  reasons  for 
the  manner  of  solving  a  problem. 

11.  A  "Prlncijjle  is  a  general  truth  upon  which  a  proc- 
ess of  computation  is  founded. 

12,  An  Example  is  a  problem  used  to  illustrate  a  prin- 
ciple, or  to  explain  a  method  of  computation. 

V  13.  An  Analysis  is  a  statement  of  the  different  steps  in 
a  solution. 

14.  A  ^ule  is  a  brief  direction  for  performing  any  com- 
putation. ,, 
Note. — These  general  definitions  apply  to  all  classes  of  numbers. 


10  INTEGERS. 

SECTION   IL 

JVOTATIOjY  AJVD   jYZTMB^ATIOJSr, 

l^  15.  JVoiaiiori  in  arithmetic  is  WBtKSSit-  expressing 
numbers  bjr  ton  charac^ero,  oallod  figmpoo. 
These  figures  are 

0123Jf56789 

called.      nmcgJit,  one,     two,  three,  four,  five,     six,    seven,  eight,  nine. 

The  figure  0,  also  called  Cipher,  denotes  nothing,  or  the 
absence  of  number ;  and  the  other  figures  represent  the 
first  nine  integers,  and  are  sometimes  called  Digits. 
y  16.  JVume7"aH07i  is  tho  art   of  reading   numbers  ^«« 

pressed  -by^^g'^geS.       (See  Manual,  page  5.) 

To  express  numbers  greater  than  9,  two  or  more  of  the 
ten  figures  must  be  combined. 

17.  In  writing  numbers,  every  ten  ones  taken  together 
are  called  a  ten. 

Ten  is  written  10 

2  tens  are  called  twenty.,  written  20 

5  tens        "  fifty,  "  50 

8  tens        "  eighty,  "  80 

9  tens        "  ninety,  "  90 

When  two  figures  are  written  together  to  express  a  num- 
ber, the  left-hand  figure  expresses  tens,  and  the  right-hand 
figure  ones.     Thus, 

Sixteen      consists  of  1  ten  and  6  ones,  written  16 
Thirty-five      "  3  tens  "    5  ones,       "       35 

Seventy-two  "  7  tens  "    2  ones,       "       72 

Ninety  "  9  tens  "    0  ones,       "       90 

18.  Every  10  tens  taken  together  are  called  a  hundred. 

One     hundred  is  written     100 

Two    hundred  "  200 

Seven  hundred  "  700 

r. 


NOTATION     AND     NUMERATION.  11 

When  three  figures  are  written  together  to  express  a 
number,  the  left-hand  figure  expresses  hundreds,  and  the 
other  two  figures  express  tens  and  ones.     Thus, 

Four  hundred  twenty-seven  consists  of  4  hundreds 
2  tens  and  7  ones,  and  is  written  Jf27 

2  hundreds  5  tens  and  6  ones,  or  two  hundred  fifty- 
six,  is  written  256 

7  hundreds  1  ten  and  8   ones,   or  seven  hundred 
eighteen,  is  written  718 

5  hundreds  3  tens  and  9  ones,  or  five  hundred  thirty- 
nine,  is  written  539 

4  hundreds  6  tens  and  0  ones,  or  four  hundred  sixty, 
is  written  4^0 

1  hundred  0  tens  and  5  ones,  or  one  hundred  five, 
is  written  ^05 

EXEJtCISES. 

1.  Write  in  words,  10,  30,  70,  23,  99,  16,  11,  12. 

2.  Write  in  words,  100,  400,  700,  350,  280,  190. 

3.  Write  in  words,  596,  281,  694,  375,  333,  899. 

4.  Write  108,  904,  301,  707,  510,  811,  600,  150. 
Express  by  figures  the  following  numbers : 

5.  Fifty,  ninety,  forty-one,  sixtj^-six. 

6.  Fourteen,  one  hundred,  four  hundred,  six  hundred. 

7.  Two  hundred  sixty,  five  hundred  ninety. 

8.  Seven  hundred  ten,  three  hundred  twenty-six. 

9.  Five  hundred  eighty-one,  six  hundred  fifteen. 

10.  Two  hundred  four,  five  hundred  three. 

11.  Seven  hundred  six,  eight  hundred  one. 

12.  Six  hundred  fifty,  seven  hundred  tw^elve. 

13.  Five  hundred  sixty-three,  two  hundred  ninety. 

14.  One  hundred  nineteen,  nine  hundred  ninety-nine. 

19.  In  writing  numbers,  every  10  hundreds  taken  to- 
gether are  called  a  thousand,  every  10  thousands  taten 
together  are  called  a  ten-thousand,  and  every  10  ten- 
thousands  are  called  a  hundred-thousand. 

When  a  figui'e  stands  at  the  left  of  hundreds  in  a  num- 
ber, it  express  thousands  ;  when  at  the  left  of  thousands,  it 


12 


INTEGERS. 


expresses  ten-thousands  ;   and  when   at  the  left   of  ten 
thousands,  it  expresses  hundred-thousands. 

One   thousand  is  written 

Five  thousand  " 

Nine  thousand  " 

Ten    thousand  " 

2  ten-thousands,  or  twenty  thousand,  " 

Sixty  thousand  " 

One  hundred  thousand  " 

Three  hundred  thousand  " 

Eight  hundred  thousand  " 

20.  Every  three  figures  in  any  number, 
counting  from  the  right,  are  called  a  Period. 
Periods  of  figures  are  separated  from  each 
other  by  commas. 

The  first,  or  right-hand  period,  consists       \ 
of  ones,  tens,  and  hundreds  ;  and  the  second 
period  of  ones,  tens,  and  hundreds  of  thousands. 
Nine  thousand  one  hundred  is  written 

Six  thousand  two  hundred  fifty  " 

Two  thousand  seven  hundred  forty-two  " 

Ten  thousand  four  hundred  " 

Fifty-six  thousand  six  hundred  seventy  " 

Nineteen  thousand  one  hundred  thirty-six     " 
Forty  thousand  seven  hundred  nine  " 

Eighty-two  thousand  six  " 

Three  hundred  seventy-six  thousand 
Five  hundred  ten  thousand 
Six  hundred  thousand  five  hundred 
Three  hundred  fifty-two  thousand  seven 
hundred  eighty-two 

EXEItCISES. 

15.  Read  5,000;  4,200;  1,3G0;  G,384;  3,569;  8,113. 

16.  Read  9,011;  5,608;  3,008;  1,040;  4,076. 

17.  Read  30,000;  57,000;  42,300;  05,850;  83,294. 

18.  Read  15,203;  47,056;  50,912;  90,052;  89,005. 

19.  Read  80,000;  25,030;  60,200;  40,475;  30,800;  55,703. 


1,000 

5,000 

9,000 

10,000 

20,000 

60,000 

100,000 

300,000 

800,000 


752,194 


5  8 


9,100 

6,250 

2,71^2 

10,400 

56,670 

19,136 

1^0,709 

82,006 

376,000 

510,000 

600,500 

352,782 


NOTATION     AND     NUMERATION.  13 

Write  the  following  numbers : 

20.  Two  thousand  ;  seven  thousand  live  liundred. 

21.  Four  thousand  one  hundred  sixty. 

22.  Nine  thousand  six  hundred  fifty-three. 

23.  Three  thousand  eight  hundred  eleven. 

24.  Seven  thousand  forty- one. 

35.  One  thousand  one ;  two  thousand  fifty. 

36.  Five  thousand  four  hundred  nine. 

37.  Sixteen  thousand  five  hundred. 

28.  Eighty-one  thousand  two  hundred  seventy. 

29.  Eleven  thousand  nine  hundred  eighty-five. 

30.  Read  275,000  ;  100,000  ;  860,000  ;  493,600  ;  815,350. 
Write  the  following  numbers  : 

81.  Two  hundred  thousand. 

32.  Six  hundred  fifty  thousand  eight  hundred. 

33.  One  hundred  nine  thousand  seven  hundred  twenty-six. 

34.  One  hundred  five  thousand  eighty.      See  Manual. 

21.  The  third  period  of  figures  con-        |  „•       |  | 
sists  of  ones,  tens,  and  hundreds  of  mill-        °  •-        o  " 

**^^s-  ^  45  9,20  8,103 

In  any  full  period  the  riglit-hand  I  :  •  -s  •  :  •!  •  • 
figure  is  ones,  the  middle  figure  tens,  %  i  i  I  s  I  I  =  i 
and    the    left-hand   figure  hundreds. 

Thus,  in  any  number  consisting  of  three  full  periods,  there 
are  ones,  ones  of  thousands,  and  ones  of  millions  ;  tens, 
tens  of  thousands,  and  tens  of  millions  ;   and  hundreds, 
hundreds  of  thousands,  and  hundreds  of  millions. 
One  million  two  hundred  thirty-one  thou- 
sand three  hundred  sixty-four  is  written        l,S31,36Jf. 
Twenty-five  million  "  25,000,000 

Nine  hundred  million  "  900,000,000 

Four  hundred  six  million  "  ^06,000,000 

EXERCISES. 

35.  Read  4,000,000;  80,000,000;  73,000,000;  9,721,312. 

36.  Read  18,271,100;  300,000,000 ;  253,729,594 ;  604,000,000. 
Write  the  following  numbers  : 

37.  Nine  million ;  fourteen  million. 


14  INTEGERS. 

38.  Four  hundred  fifty-two  million. 

39.  Nine  hundred  one  million. 

40.  Three  hundred  million  two  hundred  sixty-five  thousand. 

41.  Five  hundred  nine  million  six  hundred  twelve  thousand  nine 
hundred  eighty-five. 

22.  The  first  period  is  caU'ed  the  period  of  ones  or  units, 
the  second  the  period  of  thousands,  and  the  third  the 
period  of  millions. 

The  fourth  period  is  that  of  billions,  the  fifth  that  of 
trillions,  and  the  sixth  that  of  quadrillions. 


493,307,508,210,064,119 

I    :    :    I   :   :    I    :   •'    I    :    :    I    :   :    "i    :   : 

c     S     S        e     ?'     S        c     "     S       "2     ■»     S       "So!"       "2     =■     " 
5c2        5^2        Sc^        Cc**        Bell        o2« 

.S-2§      .5^3      J^§      J^§      jIg      Jl§ 
EXER  CIS  JE  S. 

42.  Read  4,359,006,110;  19,000,000,000;  40,060,139,194. 

43.  Read  5,236,481,279;  10,500,600,000;  92,675,244,000. 

44.  Read  3,000,000,000,000,000  ;  396,728,136,294. 

45.  Read  17,252,005,030;  18,000,039;  410,000,060,000. 
Write  the  following  numbers : 

46.  Five  billion  two  hundred  million  twenty-two  thousand  eight. 

47.  Forty-five  billion  one  hundred  fifteen  million  one  hundred 
sixty-four  thousand  eighty-nine. 

48.  Fifty-two  trillion. 

49.  One  hundred  nine  quadrillion. 

50.  Nine  billion  three  hundred  six  thousand. 

51.  Four  hundred' seventy-eight  quadrillion  two  hundred  thirty- 
four  trillion  eight  billion  five  hundred  sixteen  million  seven  hun- 
dred thousand  five  hundred  eight. 

52.  Six  hundred  nineteen  million  thirty. 

23.  The  ones,  tens,  hundreds,  thou-  ^  c  c    c  c  c    ^  ^  c 

sands,  etc.,  of  any  number  are  called  111    III    ||| 

units  of  different  orders;  ones  being  ^  ^  ^    s  si  a    ss.2 

simple  units,  or  units  of  the  first  order  ;  593,298,756 


NOTATION     AND   NUMERATION.  15 

tens,  units  of  tlie  second  order  ;  hundreds,  units  of  the  third 
order,  and  so  on. 

24 •  Every  figure  has  an  absolute  or  simple  value,  and  a 
local  value.  Its  simple  value  is  the  number  of  ones  it  ex- 
presses when  taken  alone.  Its  local  value  is  the  order  of 
units  it  expresses  in  a  number.  Thus,  8  when  taken  alone 
expresses  8  things,  8  ones,  or  8  simple  units  ;  but  when 
taken  with  other  figures  it  expresses  different  units,  accord- 
ing to  its  place.  In  80,  it  expresses  8  tens  ;  in  800,  8  hun- 
dreds ;  in  8,000,  8  thousands,  and  so  on.   (Sce  Manual,  page  5.) 

25.  A  unit  of  any  order  is  ten  times  as  great  in  value  as 
a  unit  of  the  next  lower  order.  Thus,  a  ten  is  10  times  a 
one  ;  a  hundred  10  times  a  ten  ;  a  thousand  10  times  a 
hundred,  and  so  on,  as  shown  in  the  following 

NOTATION   AND   NUMERATION    TABLE. 


10  ones 

are  1  ten, 

10  tens 

"    1  hundred, 

10  hundreds 

"   1  thousand, 

10  thousands 

"   1  ten-thousand, 

10  ten-thousands 

"   1  hundred-thousand, 

10  hundred-thousands 

"    1  milUon, 

and  so  on. 

Iten 

is  10  ones, 

1  hundred 

"  10  tens, 

1  thousand 

"  10  hundreds, 

1  ten-thousand 

"  10  thousands, 

1  hundred-thousand 

"  10  ten-thousands, 

1  million 

"  10  hundred-thousands, 

and  so  on. 

26.  ^rinc/ples  of  JVotation. 

I.  The  values  of  the  different  places  in  a  number  increase 
from  right  to  left  in  a  tenfold  ratio, 

n.  The  place  ivhich  any  figure  occupies  in  a  number  deter- 
mines the  value  expressed  by  it  in  that  number. 

III.  The  highest  period  of  any  number  must  stand  ai  the  left, 
and  tJie  succeeding  periods  in  their  order. 


16  INTEGERS. 

rV.  Every  full  period  must  consist  of  three  figures, — hun- 
dredSy  tens,  and  ones  ;  the  place  of  any  unit  not  named  in  the 
given  number  being  filled  by  a  cipher. 

Y.  Tlie  three  places  of  any  period  not  named  in  a  number 
must  be  filled  by  three  ciphers. 

27.    'Principles  of  A''iejne7^aiio7i, 

I.  Every  integer  consisting  of  more  than  three  figures  should 
be  separated  into  periods. 

II.  Each  period  of  an  integer  is  read  separately,  as  hundreds, 
tens,  and  ones  ;  the  name  of  the  period  being  pronounced  after 
the  ones. 

III.  In  reading  any  number,  the  names  of  places  and  periods 
filled  with  ciphers  are  omitted. 

EXEMCISBS. 

53.  Read  80;  290;  763;  409;  7,000;  2,009;  5,080. 

54.  Read  9,393;  6,500;  50,000;  83,400;  14,008;  10,086. 

55.  Read  512,094;  809,123;  559,026;  300,006.;  110,090. 
Write  the  following  numbers : 

56.  Eighty ;  three  hundred ;  nine  hundred  ten, 

57.  Fifty-five  ;  seven  hundred  sixteen  ;  four  hundred  one. 

58.  Eight  thousand ;  fifty  thousand  ;  ninety-two  thousand. 

59.  Six  hundred  twelve  thousand  one  hundred  sixty-five. 

60.  Fifteen  thousand  seventeen. 

01.  Four  hundred  thousand  fifty-six. 

62.  Sixty  million  ;  seven  hundred  million. 

63.  One  hundred  eighty-two  million  three  hundred  fifty-five 
thousand  four  hundred  eighty-eight. 

64.  Two  hundred  nine  million  eighteen  thousand  nine  hundred  ten. 

65.  Read  320,000,296  ;  200,165,000  ;  693,100,083  ;  501,080,276. 
60.  Read  433,279,187,695  ;  309,400,060,009. 

67.  Read  393,000,000,000,0.00,000 ;  117,371,545,903. 
Write  the  following  numbers : 

68.  Sixteen  trillion  three  hundred  ninety-six  billion. 

69.  Two  hundred  forty-seven  billion  fifty-six  thousand. 

70.  Seventy-one  trillion  two  hundred  forty-one. 

71.  Two  hundred  sixty  seven  quintillion. 


ADDITION.  17 

SECTION  III. 

INDUCXIOI^   J^^Ty   DEE^INITIONS. 

28»  1.  MxVRiA  had  3  peaches,  and  George  gave  her  4  more. 
How  many  peaches  had  she  then  ? 

2.  Frank  has  5  large  rabbits  and  6  small  ones.  How  many 
rabbits  has  he  ? 

3.  How  many  apples  are  6  apples,  4  apples,  and  7  apples  ? 

4.  How  many  birds  are  5  birds,  7  birds,  3  birds,  and  0  birds  ? 

5.  Ella  has  5  roses,  Mary  has  8,  Olive  has  4,  Alice  has  7,  Louise 
has  9,  and  Flora  has  6.     How  many  roses  have  all  the  girls  ? 

V    29.  jlddiliO?i  is  'iba  prooooo  ef  uniting  two  or  more 
numbers  to  form  one  number. 

30.  The  Amotl7Zt  or  Su7?i  is  the  result  obtained  by- 
Addition. 

31.  The  ^ai'ts  are  the  numbers  which  are  united  to 
form  the  sum. 

32.  The  Sig7i  of  Additlo?iy  made  thus  +,  is  called 
Plus  ;  and  when  written  between  numbers,  it  signifies  that 
they  are  to  be  added. 

33.  The  St(/?i  of  J^quality,  made  thus  =,  when  writ- 
ton  between  numbers  or  sets  of  numbers,  signifies  that  they 
are  equal  to  each  other.    Thus,  4  +  5  =  9;  16  ==3  +  7  + 6. 

Note.— A  number  with  the  sign   $  before  it  expresses  dollars. 

6.  What  is  the  sum  of  5  cents,  9  cents,  and  8  cents  ? 

7.  Add  9,  and  5,  and  3,  and  4,  and  7. 

8.  Add  6  books,  8  books,  5  books,  4  books,  and  9  books. 

9.  12  days  +  3  days  +  7  days  +  1  day  =  how  many  days  ? 

10.  What  is  the  amount  of  5  pens,  11  pens,  8  pens,  and  2  pens? 

11.  15  pictures  +  7  pictures  +  3  pictures  +  8  pictures  +  9  pic- 
tures =  how  many  pictures  ? 

12.  The  parts  are  12,  7,  4,  1,  5,  and  8.     What  is  the  sum  ? 


18 


INTEGERS. 


34(     ADDITION     TABLE. 


nJO    123456789 

"jo    000000000 

013    3    456789 

k(01    33456    789 

»^|555555555    5 

5    6    7    8    9  10  11  13  13  14 

,(0    123456789 

^U    1    1    1    1    1    1    1    1    1 

123456789  10 

/jjO    123456789 

"16    666666666 

6    7    8    9  10  11  13  13  14  15 

OJ0123456789 

'^\2    222222223 
23456    789  10  11 

lyjO    133456789 

'17777777777 
7    8    9  10  11  13  13  14  15  16 

qjO    123456789 

•^13    333333333 

3    4    5    6    7    8    9  10  11  12 

QJO    123456789 

0]8    888888888 

8    9  10  11  13  13  14  15  16  17 

/LJO    123456789 

■^14    444444444 
4    5    6    7    8    9  10  11  13  13 

q(01234    5    6789 

^1  9999999999 

9  10  11  13  13  14  15  16  17  18 

C^SE     I. 
The  sum  of  all  the  figures  of  any  place  not  more  than  9. 

35.  We  can  add  apples  to  apples,  dollars  io  dollars,  pens 
to  pens,  or  hours  to  hours  ;  but  we  can  not  add  apples  to 
dollars,  nor  pens  to  hours.  For  4  apples  +  9  dollars  = 
neither  13  apples  nor  13  dollars. 

Again,  we  can  add  ones  to  ones,  tens  to  tens,  or  hun- 
dreds to  hundreds  ;  but  we  can  not  add  ones  to  hundreds, 
nor  tens  to  thousands.  For  4  tens  +  9  thousands  == 
neither  13  tens  nor  13  thousands. 


36.  Example.  What  is  the  sum  of  4,216,   3,152,  and  1,321  ? 

Explanation. — Since  we  must  add  ones  to 
ones,  tens  to  tens,  hundreds  to  hundreds,  etc.,  ^°^^"^^; 
it  is  most  convenient  to  write  the  parts  with 
like  orders  of  units  in  the  same  column.  We 
then  add  each  column  separately,  writing  the 
sum  directly  under  the  column  added.  The 
sum  of  the  ones,  1  -f  2  +  6,  is  9 ;  the  sum 


Paris. 
8G8  0      Sum. 

of  the  tens. 


ADDITION.  19 

2  +  5  +  1,  is  8  ;  tlio  sum  of  the  hundreds,  3  +  1  +  2,  is 
G  ;  and  the  sum  of  the  thousands,  1  +  3  +  4,  is  8.     The 
result,  8,689,  is  the  sum  required,     csee  Manual.) 
In  this  manner  solve  and  explain  the  following 

FM  OBJuE3rS. 

Find  the  sum  in  each  of  the  first  five  problems  : 

(1)  (2)  (3)  (4)  (5) 

54  71  556  3,615  215,124 

35  26  ^  2,371  583,643 

6.  A  man  gave  $22  for  a  coat,  and  $25  for  an  overcoat.  How 
much  did  he  pay  for  both  ?  $jf,7. 

7.  A  gentleman  paid  $125  for  a  horse,  and  $163  for  a  carriage. 
How  much  did  both  cost  him  ?  $287. 

8.  A  farmer  has  14  cows,  11  oxen,  and  23  yoimg  cattle.  How 
many  head  of  cattle  has  he  ?  J^S. 

9.  One  day  a  miller  sold  321  barrels  of  flour,  the  next  day  143 
barrels,  and  the  third  day  235  barrels.  How  much  flour  did  he 
sell  in  the  three  days  ?  699  larrels. 

10.  What  is  the  sum  of  $5,418,  $51,  and  $430  ? 

11.  Add  13,300  miles,  2,051  miles,  1,435  miles,  and  3,013  miles. 
13.  la  the  Congressional  Library  at  Washington  there  are  50,700 

volumes,  and  in  the  library  of  the  Smithsonian  Institute  35,000 
volumes.     How  many  volumes  in  both  libraries  ?  75, 700. 

13.  Three  men  together  purchase  a  vessel,  A  paying  $11,735, 
B,  $10,050,  and  C,  $8,130.     What  is  the  cost  of  the  vessel  ? 

$29,895. 

14.  One  day  a  produce  dealer  bought  from  three  men  730  bush- 
els, 145  bushels,  and  1,134  bushels  of  oats.  How  many  oats  did  he 
buy  ?  1^989  tmshels. 

15.  An  army  containing  41,430  men  received  two  reinforcements, 
the  first  of  13,325  men,  and  the  second  of  34,334  men.  How  many 
men  were  then  in  the  army  ?  78,889. 

16.  A  dealer  in  real  estate  sold  three  city  lots  for  $1,220  each, 
another  lot  for  $3,135,  and  a  farm  for  $12,310.  For  how  much  did 
they  sell?  $17,995. 


2.J  INTEGERS. 

Cj?4.SiG     II. 
The  sum  of  all  the  figures  of  any  place  more  than  9. 

37.  5  +  8  +  3  =  16,  and  16  =  1  ten  and  6  ones. 

5  hundreds  +  8  hundreds  +  3  hundreds  =  16  hundreds, 
and  16  hundreds  =  1  thousand  and  6  hundreds. 

7  tens  +  9  tens  +  8  tens  =  24  tens,  and  24  tens  =  2  hun- 
dreds and  4  tens.     Hence 

When  the  sum  of  the  units  of  any  order  exceeds  0,  the  tens 
of  this  sum  are  units  of  the  next  higher  order. 

38,  Ex.  "What  is  the  sum  of  3,475,  2,694,  and  1,383? 
ExPL:iNATioN.  ■ —  We   write   the   parts  as  in 

Case  I.,  draw  two  horizontal  lines  underneath,     ^^^^^^  solution. 
as  shown  in  the  Fkst  Sohition,  and  then  add.         3Jf75 
The  sum  of  the  ones,  3  +  4  +  5,  is  12,  or  2        ^^^| 

ones  and  1  ten.     We  write  the  2  ones  below        — -^ 

the  lower  line  as  the  ones  of  the  sum,  and  the 
1  ten  in  tens'  place,  between  the  two  lines,  to         7532 
bo  added  with  the  column  of  the  tens.     The 
sum  of  the  tens,  1  +  6  +  9  +  7,  is  23,  or  8  tens  and  2 
hundreds.     We  write  the  3  tens  as  the  tens  of  the  sum,  and 
the  2  hundreds  in  hundreds'  place  between  the  two  lines. 
The  sum  of  the  hundreds,  2  +  3  +  6  +  4,  is  15,  or  5  hun- 
dreds and  1  thousand.     We  write  the  5  himdreds  as  the 
hundreds  of  the   sum,  and  the  1  thousand  in  thousands' 
place  between  the  lines.     The  sum  of  the  thousands,  1  +  1 
+  2  +  3,  is  7,  and  this  we  write  as  the  thousands  of  the 
sum.     The  result,  7,532,  is  the  sum  required. 

Explanation. — In    the   Second  Solution  we 
write  the  parts  as  before,  draw  one  horizontal  second  soi.utiox. 
Iin3  underneath,  and  then  add.     The  sum  of        3Ji.75 
the  ones  is  12,  or  2  ones  and  1  ton.    We  write         i&ni 

the  2  ones  below  the  line  for  the  ones  of  the         

sum,  and  add  the  1  ten  with  the  column  of         7o3^ 
tens.     The  sum  of  all  the  tens  is  23,  or  3  tans 


ADDITION 


21 


and  2  limidreds.  We  write  the  3  tens  as  the  tens  of  the 
sum,  and  add  the  2  hundreds  with  the  column  of  hun- 
dreds. The  sum  of  all  the  hundreds  is  15,  or  5  hundreds 
and  1  thousand.  We  write  the  5  hundreds  as  the  hundreds 
of  the  sum,  and  add  the  1  thousand  with  the  column  of 
thousands.  The  sum  of  all  the  thousands  is  7,  and  this  wo 
write  as  the  thousands  of  the  sum.  The  result,  7,532,  is  the 
sum  required. 


JPJt  OB  LJ^31S. 

17.  The  numbers  on  these  packages 
of  express  freight  show  their  weight 
in  pounds.  What  is  the  weight  of  the 
marked  packages  that  have  been  un- 
loaded ?  971  pounds. 

18.  How  many  pounds   do   the   cask,   barrel,   and    half-barrel 
weigh  ?  1, 133. 

19.  What  is  the  weight  of  all  the  marked  boxes  on  the  express 
wagon?  Jj.,  110  pounds. 

20.  How  much  does  all  the  marked  freight  on  the  wagon  weigh  ? 
SI.  What  is  the  total  weight  of  all  the  marked  packages  shoM^n 

in  the  x^icture  ?  5,373  pounds. 

22.  In  the  Old  Testament  arc  39  "books,  and  in  the  New  Testa- 
ment 27  books.     How  many  books  are  in  the  Bible  ?  60. 

23.  A  cabinet-maker  paid  $125  for  black-walnut  lumber,  and  $90 
for  mahorran-.^     How  much  did  the  lumbcT  cost  him  ?  $215. 


22  INTEGERS. 

24.  A  merchant  taiJor  bought  three  pieces  of  broadcloth,  the 
first  containing  27  yards,  the  second  45  yards,  and  the  third  84 
yards.     How  many  yards  did  he  buy  ? 

25.  One  day  a  man  traveled  241  miles  by  railroad,  57  by  steam- 
boat, and  14  by  stage.     How  far  did  he  travel  ?  312  miles. 

26.  Henry  is  16  years  old,  his  father  is  29  years  older  than  he, 
and  his  grandfather  32  years  older  than  his  father.  How^  old  is  his 
grandfather  ?  r^^  years. 

27.  From  the  creation  of  the  world  to  the  Christian  era  were 
4,004  years.  How  many  years  from  the  creation  to  the  end  of  the 
present  year  ? 

28.  A  fruit  dealer  bought  56  barrels  of  russet  apples,  74  barrels 
of  pippins,  69  barrels  of  spitzenbergs,  and  83  barrels  of  greenings. 
How  many  apples  did  he  buy  ?  ^82  darrcls. 

39.  The  facts  deduced  in  Arts.  35  and  38  maybe  stated  as 

Principles  of  Addiiioti, 

I     I.  Only  abstract  numbers  or  like  conerete  numbers  can  be 
\  added. 
y  I     II.  Only  like  orders  of  units  in  different  numbers  can  be 
;  added. 

i     m.  WJieyi  the  sum  of  the  units  of  any  order  exceeds  0,  the 
I  tens  of  this  sum  are  units  of  the  next  higher  order. 

40.  Upon  these  principles  is  based  the 

(Rule  for  Addiiioji   of  Tnteffei's, 

I.  Add  like  orders  of  units,  and  icriie  the  ones  of  the  sum  in 
the  result. 

H.  Add  the  tens  of  the  sum  of  any  order  loiih  the  next  higher 
order. 

in.  Write  the  whole  sum  of  the  highest  order  of  units  given. 

Note. — Since  the  tens  of  the  sum  of  any  column  must  be  added  -with  the 
next  left-hand  column,  it  is  iu  general  more  convenient  to  commence  at 
the  rirrht  to  add. 


ADDITION.  23 


rjz  or,  Tj  E3IS. 

29.  A  nursery-man  sold  during  the  year  3,729  apple-trees,  1,415 
pear  trees,  974  peacli  trees,  567  plum-trees,  918  cherry-trees,  and 
1,584  ornamental  trees.     How  many  trees  did  he  sell  ?         9,187. 

30.  One  year  a  farmer  raised  649  bushels  of  oats,  422  bushels  of 
corn,  178  bushels  of  wheat,  and  96  bushels  of  barley.  How  much 
grain  did  he  raise  ?  l^SJ^B  bushels. 

31.  A  merchant  pays  his  book-keeper  $1,250  a  year,  two  clerks 
$825  each,  and  a  boy  $175.  How  much  do  their  salaries  amount 
to?  $3,075. 

32.  An  Erie  canal-boat  has  on  board  273  barrels  of  flour  for 
Utica,  385  barrels  for  Albany,  and  465  barrels  for  New  York.  Hov/ 
much  flour  has  she  on  board  ?  1,123  barrels. 

33.  A  grocer,  in  purchasing  his  first  stock,  paid  $466  for  sugai-s, 
$387  for  syrups,  $196  for  teas,  and  $1,760  for  other  goods.  What 
was  the  cost  of  his  stock  ?       ^  $2,809. 

34.  At  an  auction  a  woman  bid  off  one  carpet  for  $24,  anothei- 
for  $36,  some  oil-cloth  for  $7,  and  some  window-shades  and  fixtures 
for  $12.     What  was  the  amount  of  her  bill  ?  ^7^ 

35.  At  the  New  York  Cattle  Market  the  number  of  beeves  on 
sale  last  Tuesday  was  as  follows  :  number  remaining  over  from 
Monday,  396;  received  by  Erie  Railroad,  1,516;  by  Hudson  River 
Railroad,  1,044  ;  by  Harlem  Railroad,  1,185 ;  by  Camden  &  Amboy 
Railroad,  296  ;  by  Hudson  River  boats,  210  ;  by  New  Jersey  Cen- 
tral Railroad,  329 ;  and  on  foot,  311.  How  many  beeves  were  on 
sale?  5,287. 

36.  A  man  built  a  house,  which  cost,  for  brick  and  stone,  $375 ; 
for  lumber,  $540 ;  for  other  materials,  $224  ;  for  excavation,  $72 ; 
for  mason  work,  $284 ;  for  carpenter  work,  $580  ;  and  for  painting, 
glazing,  and  paper  hanging,  $225.  How  much  did  the  house  cost 
him?  ■  $2,300. 

37.  At  one  time  the  rolling  stock  of  the  New  York  Central  Rail- 
road was  211  locomotives,  196  first-class  passenger  cars,  41  second- 
class  and  emigrant  cars,  61  baggage,  mail,  and  express  cars,  2,760 
freight  cars,  and  350  gravel  cars.  What  was  the  whole  number  of 
cars?  3,619. 


24  I  X  T  E  G  E  R  S. 

88.  I  paid  $325  for  a  lot,  |1,42G  for  building  a  house  upon  it„ 
$589  for  building  a  barn  and  carriage  house,  $74  for  fences,  anc' 
$48  for  grading  the  lot.  For  liow  much  must  I  sell  the  property 
to  gain  $338  ?  $2,800. 

39.  On  Monday  morning  a  merchant  had  $1,767  in  the  bank. 
That  day  he  deposited  $94;  on  Tuesday,  $118;  on  Wednesday, 
$78;  on  Thursday,  $141  ;  on  Friday,  $52  ;  and  on  Saturday,  $279. 
How  much  had  he  on  deposit  at  the  end  of  the  week  ?       $2,524. 

40.  Find  the  sum  of  15  million  9  thousand  17,  9  million  503, 
675  thousand  899,  and  245  million  320  thousand  8. 

270  million  11  tlioiisand  J^32. 

41.  In  each  of  the  two  following  sets  of  numbers,  find  the  sunn 
of  all  the  numbers  above  e. 

X     21,365_^  42.  From  a  to  d.  2,194,756_ 

43.  From  I  to  e.  40,373,254 


18,890" 

—a 
-h 

54,363" 

— /! 

27,5411^ 

53,027 

—e 

34,198 
44,254" 

-f 

87,079 

-0 

-h 

73,250~ 

—  i 

19,000 

48,408 

-k 

91,510 

-I 

60,009 

38,482~^ 

~-7i 

19,504 

—0 

05,587 

28,385 

-P 

-<1 

-r 

y    78,126 

44.  Above/.  90,000,383 

45.  From  a  to  g.  0,275,851 


—b 
— c 
—d 


46.  From  d  to  i.  12,593 

47.  From  h  to  m.  6,005     ^ 


48.  Below  «.  373,58^ 


49.  Eclow  h.  218     ^ 


0.  From /to  7.7.  1,694,583 


51.  From  g  to  o.  657,679_ 

-I 


-J 
53.  From  e  to  I.  73,418  ~ 


52.  From,;  to  i7.  500,000,290 


54.  From  a  to  ;.  1,547 

55.  Above  i.  4,293,500_^ 

56.  Below?.  400,000     '^ 

57.  From/ to  72.  44^ 

58.  From  Ic  to  r.         '  68,974_^ 

59.  From  d  to  p.  13,987,457     ^ 

60.  What  is  thc-Gum  of  the  two  answers  of  Problem  41  ? 

61.  The  sum  of  the  two  answers  of  Problem  42  ? 

62.  The  sum  of  the  four  answers  of  Problems  43  and  44  ? 

63.  Add  the  answers  of  Problems  45,  46,  47,  and  43, 

64.  Add  the  answers  of  Problems  49,  50,  51. 

65.  Add  the  answers  of  Problems  52,  53,  54. 

66.  Add  the  answers  of  Problems  55,  56,  57,  and  58. 


ADDITION.  ^  1^ 

67.  What  is  the  sum  of  the  two  answers  of  Problem  S^^^^^^ifl^-S^^ 

See  Manual. 

68.  How  many  rods  of  fence  will  it  take  to  inclose  a  field  that  is 
^8  rods  long  on  each  side,  and  29  rods  wide  on  each  end  ?      i<?^. 

69.  I  have  a  farm  176  rods  long  and  115  rods  wide.  A  fence  ex- 
tends around  it ;  3  inside  fences  extend  from  end  to  end ;  and  4 
other  fences  from  side  to  side.  How  many  rods  of  fence  on  the 
farm?  1,570. 

70.  Thomas  Jefferson  was  born  a.d.  1743,  and  lived  to  be  83 
years  old.     In  what  year  did  he  die  ? 

71.  A  schooner  cleared  from  Chicago  for  Buffalo,  having  on 
board  14,397  bushels  of  wheat,  5,810  bushels  of  corn,  and  2,118 
bushels  of  oats.     How  much  grain  had  she  in  her  cargo  ? 

72.  A  grocer  bought  three  hogsheads  of  sugar,  containing  1,467 
pounds,  1,324  pounds,  and  1,296  pounds;  also,  two  barrels  con- 
taining 254  pounds  and  237  pounds.  How  much  sugar  did  he 
buy  ?  1^^578  'pounds. 

73.  A  man  bought  a  farm,  paying  $2,375  down.  After  making 
three  other  payments  of  $1,148,  $1,096,  and  $1,260,  the  amount  un- 
paid was  $5,896.     What  was  the  cost  of  the  farm  ? 

74.  How  many  days  in  the  first  six  months  of  the  year,  January, 
March,  and  May  each  having  31  days,  April  and  June  each  SO 
days,  and  February  28  days  ? 

75.  At  the  battle  of  Fort  Donelson,  the  Union  loss  was  44G 
killed,  1,735  wounded,  and  150  taken  prisoners.  The  Confederate 
loss  was  237  killed,  1,007  wounded,  and  13,300  taken  prisoners. 
What  was  the  whole  number  killed  ?  The  whole  number  wounded  ? 
The  whole  number  taken  prisoners  ? 

Killed,  683  ;  wounded,  2,71^.2  ;  prisoners^  13,450. 

76.  What  was  the  total  loss  to  each  army  ? 

Union,  2,331;  Confederate,  UM^- 

77.  The  property  in  a  certain  school  district  is  assessed  as  fol- 
lows:  to  A,  $3,875;  B,  $1,050;  C,  $13,250;  D,  $600 ;  E,  $5,875 ; 
F,  $250;  Glass  Manufactory,  $105,750;  H,  $3,000;  I,  $150; 
J,  $860;  K,  $14,180;  L,  $375;  M,  $53,000;  National  Bank, 
$151,500;  O,  $13,760;  P,  $670;  Q,  $19,960;  Railroad  Company, 
$89,500;  S,  $960;  T,  $350  ;  U,  $26,675  ;  V,  $17,625;  and  W,  $275. 
What  is  the  assessed  valuation  of  the  district  ?  $523,490. 

2 


26  INTEGERS. 

SECTION  IV. 

S  VS  T^A.  C  TIOJV, 
INDUCTIONS-  ^isrr)  iDEFiisrjTioisrs. 

41*  1.  Myba  brought  8  roses  to  school,  and  gave  5  of  them  to 
her  teacher.    How  many  had  she  left  ? 

2.  In  a  garden  there  are  11  fruit-trees;  4  of  them  are  plum- 
trees,  and  the  others  cherry-trees.     How  many  are  cherry-trees  ? 

3.  If  you  buy  13  Brazil-nuts,  and  eat  8  of  them,  how  many  will 
you  have  left  ? 

4.  Ellen  had  14  books  upon  two  shelves,  and  7  of  them  were  on 
the  lower  shelf     How  many  were  on  the  upper  shelf? 

5.  Kobert  is  13  years  old,  and  Edward  is  9.  How  much  older  is 
Robert  than  Edward  ? 

6.  One  day  a  lawyer  wrote  15  letters,  writing  7  of  them  in  the 
forenoon.    How  many  did  he  write  in  the  afternoon  ? 

7.  A  laborer  received  $9  for  his  week's  work,  and  spent  $6. 
How  much  money  had  he  left  ? 

V^    42.  Subtraction  is  ifeo  proooriic  of  taking  one  of  two 
numbers  from  the  other. 

43.  The  Remainder  or  !2)iffe7'e7ice  is  the  result  ob- 
tained by  subtraction. 
sy-     44.  The  Mi7iue7id  is  that-ono  of  Ivvu  nuittlieja  from 

which  the  other  is  to  be  taken. 
.^  45.  The  Subtrahend  is  that  ono  ef  two  numb oifs.  which 
is  to  be  taken  from  the  other. 

Note. — The  subtrahend  can  never  be  a  greater  number  than  the  minuend. 

46.  The  Si^7i  of  Subtraction ^  made  thus  — ,  is 
called  Minus;  and  when  written  between  two  numbers,  it 
signifies  that  the  number  after  it  is  to  be  subtracted  from 
the  number  before  it.  Thus,  18  —  12  signifies  that  12  is  to 
be  subtracted  from  18. 

8.  15  men  —  9  men  =  how  many  men  ? 

9.  17  pencils  —  7  pencils  =  how  many  pencils  ? 


SUBTRACTION. 


27 


10.  From  14  hours  subtract  5  hours. 

11.  From  18  pounds  subtract  9  pounds.     Which  number  is  the 
minuend  ?    Which  is  the  subtrahend  ?    Which,  the  remainder  ? 

12.  The  minuend  is  16,  and  the  subtrahend  5.      What  is  the 
remainder  ? 

13.  What  is  the  difference  between  13  and  10  ? 

14.  What  is  the  remainder  when  8  is  subtracted  from  17  ? 

47.     SUBTRACTION    TABLE. 


0 

1    3 

0  0 

1  2 

3    4    5    6    7    8    9 
0    0    0    0    0    0    0 
3    4    5    6    7    8    9 

K  j  5    6    7    8    9  10  11  13  13  14 

^U    555555555 

013    3456789 

Ml 

0 

2    3 
1    1 

1    2 

4    5    6    7    8    9  10 
1111111 
3    4    5    6    7    8    9 

/>  j  6    7    8    9  10  11  13  13  14  15 

^J6    666666666 

0133456789 

2|l 

0 

3    4 
3    3 
1    3 

5    6    7    8    9  10  11 

3    3    3    3    3    3    3 
3    4    5    6    7    8    9 

« j  7    8    9  10  11  13  13  14  15  16 

(|7    777777777 
0133456789 

3]l 

4    5 
3    3 

6    7    8    9  10  11  13 
3    3    3    3    3    3    3 

ft  (  8    9  10  11  13  13  14  15  16  17 

o|8    888888888 

0 

1    3 

3    4    5    6    7    8    9 

0133456789 

Mt 

0 

5    6 

4    4 
1    3 

7    8    9  10  11  13  13 
4    4    4    4    4    4    4 
3    4    5    6    7    8    9 

Q  (  9  10  11  13  13  14  15  16  17  18 

y]9    999999999 

0133456789 

c-a.se  I. 

No  figure  of  the  Subtrahend  greater  than  the  corresponding 
figure  of  the  Minuend. 

48.  We  can  subtract  apples  from  apples,  dollars  from 
dollars,  pens  from  pens,  or  hours  from  hours  ;  but  we  can 
not  subtract  apples  from  dollars,  nor  pens  from  hours. 
For  13  apples  —  4  dollars  =  neither  9  apples  nor  9  dollars. 

Again,  we  can  subtract  ones  from  ones,  tens  from  tens, 
or  hundreds  from  hundreds  ;  but  we  can  not  subtract  ones 
from  hundreds,  nor  tens  from  thousands.  For  9  thousands 
—  4  tens  =  neither  5  tens  nor  5  thousands. 


28  INTEGERS. 

49.  Ex.  From  5,267  subtract  2,215.  solution. 

Explanation. — Since  we  must  subtract    526  7  Minuend. 
ones  from  ones,  tens  from  tens,  hundreds    22 15  Subtrahend. 
from  hundreds,  etc.,  it  is  most  convenient    SO 52  Remainder. 
to  write  the  figures  of  the  subtrahend 
under  the  figures  of  like  orders  in  the  minuend.      We 
then  subtract  each  figure  of  the  subtrahend  from  the  figure 
above  it  in  the  minuend,  writing  the  result  directly  below 
in  the  remainder.     5  ones  from  7  ones  leave  2  ones  ;  1  ten 
from  6  tens  leave  5  tens;   2  hundreds  from  2  hundreds 
leave  0  hundreds  ;  and  2  thousands  from  5  thousands  leave 
3  thousands.    The  result,  3,052,  is  the  difference  or  remain- 
der required. 

In  the  same  manner  solve  and  explain  the  following 

PB  OBZEMS. 

(1)        (3)  (3)  (4)  (6) 

From  85        459        4978        13379  feet         $2384 

Subtract     43        348        3264  3148  feet.  1073 

6.  From  a  chest  of  tea,  which  contained  76  pounds,  a  grocer  sold 
43  pounds.     How  much  tea  remained  in  the  chest  ?      S3  pounds. 

7.  From  a  flock  of  396  sheep  a  drover  bought  194.  How  many 
sheep  were  left  in  the  flock  ?  202. 

8.  A  man  bought  a  house  and  lot  for  $2,375,  paying  $1,225 
down.     How  much  did  he  then  owe  on  the  place  ?  $1, 150. 

9.  A  contractor  received  $7,875  for  building  a  railroad  bridge, 
and  it  cost  him  $5,450  to  build  it.     How  much  was  his  profit  ? 

10.  In  a  city  school  there  are  849  pupils,  of  whom  437  are  girls. 
How  many  are  boys  ?  ^12. 

11.  A  and  B  together  bought  a  steamboat  for  $78,385,  and  A 
furnished  $45,385  of  the  purchase-money.  How  much  did  B  fur- 
nish ?  $33,000. 

13.  From  9  million  548  thousand  276,  subtract  5  million  84 
thousand  153.  lierminder,  4,514,123. 


SUBTRACTION.  29 


C^^SJE     II. 

Any  figure  of  the  Subtrahend  greater  than  the  correspond- 
ing figure  of  the  Minuend. 

50i  If  the  minuend  is  5,  and  the  subtrahend  is  2,  the 
difference  is  3. 


5 
2 

5  +  4  =  9 
2  +  4  =  6 

5  +  7  =  12 
2  +  7=9 

3 

3 

3 

If  4  be  added  to  both  minuend  and  subtrahend,  the  dif- 
ference is  3,  as  before.  Again,  if  7  be  added  to  both  min- 
uend and  subtrahend,  the  difference  is  still  3.     Hence, 

The  difference  or  remainder  is  not  affected  by  adding  the  same 
number  to  both  minuend  and  subtrahend. 

51.  Ex.  1.  From  40,658,  subtract  21,385. 

Explanation. — "Writing  the  numbers  as  in  Case  sonjuoy. 
I.,  we  commence  at  the  right  to  subtract.  5  ones  Ji.0658 
from  8  ones  leave  3  ones,  which  we  write  as  the  ^  1^8o 
ones  of  the  remainder.  Since  we  can  not  subtract  19273 
8  (tens)  from  5  (tens),  and  since  the  difference  will 
not  be  affected  by  adding  the  same  number  to  both  minuend 
and  subtrahend  (50),  we  add  10  (tens)  to  the  5  of  the  minu- 
end, and  1  (hundred  =  10  tens)  to  the  3  of  the  subtrahend. 
Vfe  then  subtract  8  from  15,  and  4  from  6,  writing  the 
results  7  (tens)  and  2  (hundreds),  as  the  tens  and  hundreds 
of  the  remainder.  Since  we  can  not  subtract  1  (thousand) 
from  0  (thousand),  we  add  10  (thousands)  to  the  0  of  the 
minuend,  and  1  (ten-thousand  =  10  thousands)  to  the  2  of 
the  subtrahend.  We  then  subtract  1  from  10,  and  3  from 
4,  and  write  the  results,  9  (thousands)  and  1  (ten-thousand), 
as  the  thousands  and  ten-thousands  of  the  remainder.  The 
result,  19,273,  is  the  remainder  required. 


30  INTEGERS. 

Ex.  2.  From  923  subtract  48. 
Explanation. — Since  we  can  not  subtract  8  (ones)    soltttion. 
from  3  (ones),  we  add  10  (ones)  to  the  3  of  tlie      ^^^ 
minuend,  and  1  ten  ( =  10  ones)  to  the  4  of  the         "^ 
subtrahend.     Then  8  from  13  leaves  5.     Since  we      ^^^ 
can  not  subtract  5  tens  (4  +  1)  from  2  tens,  we 
add  10  (tens)  to  the  2  of  the  minuend,  and  1  (hundred  = 
10  tens)  to  the  subtrahend.     Then,  5  from  12  leaves  7,  and 
1  from  9  leaves  8.     The  result,  875,  is  the  remainder  re- 
quired. 

Ex.  3.   From  1,000  subtract  257. 

Explanation. — In  solving  this  example,  we  first  boltjtion. 
add  10  (ones)  to  the  minuend,  and  1  (ten)  to  the  1000 
subtrahend,  and  subtracting  7  from  10,  we  obtain  _l^Z 
3  ones.  "We  next  add  10  (tens)  to  the  minuend,  743 
and  1  (hundred)  to  the  subtrahend,  and  subtract- 
ing 6  from  10,  we  obtain  4  tens.  We  then  add  10  (hun- 
dreds) to  the  minuend,  and  1  (thousand)  to  the  subtrahend, 
and  subtracting  3  from  10  and  1  from  1,  we  obtain  7  hun- 
dreds and  0  thousands.     The  result,  743,  is  the  remainder 

required.        See  Manual. 

In  the  same  manner  solve  and  explain  the  following 

PB  OBLEMS. 

(13)        (14)  (15)  (16) 

From  93         416         1483  men  433150  miles 

Subtract      28         198         _C45  men.         145316  miles. 

17.  A  man  having  $725  on  deposit,  draws  out  |268.    How  much 
has  he  left  in  the  bank  ?  $Jf57. 

18.  A  merchant  sold  a  bill  of  goods  for  $173,  and  his  profits 
were  $37.     How  much  did  the  goods  cost  him  ?  $136. 

19.  A  man's  income  is  $1,675,  and  his  expenses  are  $948.    How 
much  does  he  save  ?  $727. 

20.  1,378  tons  —  985  tons  =  how  many  tons  ?  393, 

21.  1,045  bushels  —  66  bushels  =  how  many  bushels  ?       979, 


SUBTRACTION. 


31 


22.  How  much  more  does  the  heaviest  bale  of  cotton  weigh  than 
the  large  box  near  the  ship  ?  382  pounds. 

23.  How  much  more  do  the  3  cotton  bales  weigh  than  the  heav- 
iest marked  package  ? 

24.  "What  is  the  difference  between  the  weight  of  the  2  barrels 
and  the  bale  of  cotton  which  the  3  men  are  handling  ? 

25.  Which  weighs  the  most,  the  ])ale  on  which  one  of  the  men  is 
standing,  or  the  4  lightest  marked  packages,  and  how  much  ? 

26.  All  the  marked  boxes  are  to  be  shipped  on  board  the  vessel 
lying  at  the  wharf,  and  all  the  other  marked  freight  has  been 
landed  from  her.  Will  she  take  in  more  or  less  freight  than  she 
has  discharged,  and  how  much  ?  BJfS  pounds  less. 

27.  A  Boston  provision  dealer  having  1,296  barrels  of  pork,  ships 
748  barrels  to  Liverpool.    How  many  barrels  has  he  then  on  hand  ? 

28.  In  an  election  for  Member  of  Congress,  A  received  12,031 
votes,  and  B,  10,032  votes.  Which  candidate  was  elected,  and  by 
what  majority  ?  A,h/  a  majority  of  1,999. 

29.  A  yoke  of  oxen,  before  being  fattened,  weighed  2,586  pounds, 
and  after  being  fattened,  3,174  pounds.  How  much  had  they 
gained  ?  588  pounds. 

30.  An  ox  weighed  1,326  pounds  on  foot,  and  996  pounds  when 
slaughtered.      What  was  the  difference  between   the    live  and 


dressed  weights 


830  pounds. 


32  INTEGERS. 

52.  The  faots  deduced  in  Arts.  48,  50,  may  be  stated  as 

l^rinciptes  of  Subtraction, 

I.  Only  abstract  numbers  or  like  concrete  numbers  can  be 
subtracted  the  one  from  the  other. 

II.  Only  like  orders  of  units  can  be  subtracted  the  one  from 
the  other. 

m.  The  difference  or  remainder  is  not  affected  by  adding 
the  same  number  to  both  minuend  and  subtrahend. 

53.  Upon  these  principles  is  based  the 

^ute  for  Subtraction  of  Integers* 

I.  Subtract  units  from  units  of  like  orders,  writing  each  dif- 
ference for  the  same  order  of  units  in  the  result. 

n.  Wheii  any  order  of  units  in  the  subtrahend  is  greater  in 
value  than  the  corresponding  order  in  the  minuend,  add  10 
units  of  the  same  order  to  the  minuend,  and  1  unit  of  the  next 
higher  order  to  the  subtrahend. 

Notes. — ^1.  When  no  figure  of  the  subtrahend  exceeds  in  yalue  the  cor- 
responding figure  of  the  minuend,  we  may  commence  at  the  right  or  at  the 
left  to  subtract. 

2.  "When  one  or  more  figures  of  tlie  subtrahend  exceed  in  value  the  cor- 
responding figures  of  the  minuend,  it  is  in  general  more  convenient  to 
commence  at  the  right  to  subtract.    See  Manual. 

PJt  OJiJuEMS. 

(31)           (32)  (33)  (34) 

From       3250         10000         25600  gallons         342651  reams 
Subtract      89         24  8008  gallons.        142652  reams. 

35.  A  man's  property  is  valued  at  $75,000,  of  which  $48,766  is 
in  real  estate.     How  much  is  his  personal  property  worth  ? 

86.  A  grain  buyer  in  Milwaukee  receives  an  order  for  12,500 
bushels  of  "  No.  1 "  wheat,  and  has  only  7,645  bushels  in  store. 
How  much  must  he  purchase  to  fill  the  order  ?         J!j,,855  htisheh. 

37.  A  religious  society,  after  raising  $17,675  by  subscrii^tion, 
contracted  for  the  erection  of  a  church  for  $22,500.  How  much 
remains  yet  to  be  raised  ?  $Jt,825. 


SUBTRACTION.  33 

38.  Christopher  Columbus  was  bom  a.d.  1437,  discovered  Amer- 
ica A.D.  1492,  and  died  a.d.  1506.  How  old  was  he  when  he  dis- 
covered America  ?   How  old  when  he  died  ?    65  years  ;  69  years. 

39.  26,957,239  bushels  of  salt  were  used  in  the  United  States  in 
1860,  and  of  this  amount  14,094,227  bushels  were  imported.  How 
many  bushels  were  of  home  manufacture  ?  12,863,002. 

40.  The  distance  from  Albany  to  Buffalo  is  297  miles,  and  from 
Albany  to  Rochester,  229  miles.  How  far  is  it  from  Rochester  to 
Buffalo  ?  68  miks. 

41.  At  a  saw-mill  100,000  feet  of  pine  lumber  were  sawed  in  one 
month,  and  47,250  feet  of  it  were  sold.  How  much  remained  at  the 
mill?  52,750  feet. 

42.  One  year  the  receipts  of  the  Tliird  Avenue  Railroad  of  New 
York  city  were  $564,839,  and  the  expenses  were  $307,188.  How 
much  were  the  profits  ?  $257, 651. 

Lake  Superior  has  an  elevation  of  023  feet  above  tide ;  Lake 
Huron,  of  591  feet ;  Lake  Erie,  of  565  feet ;  Lake  Ontario,  of  232 
feet ;  Great  Salt  Lake,  of  4,200  feet ;  and  Lake  Titicaca,  of  12,785 
fcet. 

43.  How  much  higher  is  Lake  Superior  than  Lake  Erie  ? 

44.  How  much  higher  is  Lake  Superior  than  Lake  Huron  ? 

45.  Great  Salt  Lake  is  how  much  higher  than  Lake  Ontario  ? 
40.  How  much  fall  is  there  in  Niagara  River  ? 

47.  Lake  Superior  is  how  much  higher  than  Lake  Ontario  ? 

48.  How  much  higher  is  Lake  Titicaca  than  Lake  Erie  ? 

49.  How  much  higher  is  Lake  Huron  than  Lake  Ontario  ? 

50.  How  much  fall  is  there  between  Lake  Huron  and  Lake  Erie  ? 

51.  Lake  Titicaca  is  how  much  higher  than  Great  Salt  Lake  ? 

52.  From  a  farm  of  417  acres,  the  owner  sold  132  acres  to  one 
man,  and  96  acres  to  another.     How  much  land  had  he  left  ? 

53.  A  bank  teller  received  a  salary  last  year  of  $1,250.  His  per- 
sonal expenses  were  $753,  and  he  bought  a  village  lot  for  $213,  and 
paid  out  $149  for  improvements  upon  it.  How  much  money  had 
he  at  the  end  of  the  year  ?  $135. 

54.  A  drover  having  319  head  of  cattle,  sold  98  head  to  one 
butcher  and  127  head  to  another.     How  many  cattle  had  he  left  ? 

2* 


3^  INTEGERS. 

55.  An  estate  worth  $35,474  is  encumbered  to  tlie  amount  of 
$17,625.     How  much  is  it  worth  above  the  incumbrance  ? 

56.  A  farmer  having  113  sheep,  sold  57  of  them,  and  afterward 
bought  83  more.     How  many  had  he  then  ?  139. 

57.  A  man  at  his  death  left  an  estate  worth  $48,765.  He  owed 
$13,596,  and  bequeathed  $12,750  to  his  widow,  $5,875  to  charitable 
institutions,  and  the  balance  to  his  only  son.  How  much  did  his 
son  receive  ?  $16^544' 

58.  A  regiment  was  mustered  into  the  service  with  976  men,  and 
afterward  received  274  recruits.  During  service  its  losses  were  38 
killed  in  battle,  94  wounded,  54  taken  prisoners,  69  discharged  for 
sickness,  13  died  from  sickness,  and  47  deserted.  Of  how  many 
men  did  the  regiment  then  consist  ?  935. 

59.  A  lumber  dealer  sold  a  quantity  of  plank  for  $746,  making  a 
profit  of  $148.     How  much  did  the  lumber  cost  him  ?  $598. 

60.  A  hardware  merchant,  who  owes  a  grocer  $113  on  account, 
sells  him  a  cook  stove  for  $32,  a  parlor  stove  for  $28,  and  some 
pipe  for  $7,  and  pays  the  balance  in  cash.  How  much  money  does 
the  grocer  receive  ? 

61.  A  clergyman  had  his  life  insured  for  $3,500.  At  the  time  of 
his  death  $376  of  his  salary  was  unpaid ;  he  owned  a  house  and  lot 
worth  $3,275,  but  upon  it  there  was  a  mortgage  for  $1,390 ;  and 
his  other  debts  amounted  to  $294.  How  much  did  he  leave  his 
family?  $5,467. 

62.  A  block  of  stores,  valued  at  $37,675,  and  goods  worth 
$69,325,  were  destroyed  by  fire.  The  buildings  were  insured  for 
$31,875,  and  the  goods  for  $49,290.  What  was  the  loss  on  the 
buildings  ? 

63.  What  was  the  loss  on  the  goods  ? 

64.  How  much  did  the  loss  on  the  goods  exceed  the  loss  on  the 
buildings?  $U,235. 

65.  June  1,  a  grocer  bought  1,754  pounds  of  ^  sugar,  1,249 
pounds  ©,  2,154  pounds  B,  1,864  pounds  C,  2,752  pounds  W.  I., 
and  1,954  pounds  N.  O.  August  1,  he  had  on  hand  967  pounds  0, 
856  pounds  ®,  1,182  pounds  B,  1,692  pounds  C,  2,158  pounds 
W.  I.,  and  369  pounds  IST.  O.  How  much  sugar  of  each  brand  had 
he  sold  in  the  month  ?      See  Manual. 


MULTIPLICATION.  35 

SECTION  V. 

MUJO  TITZIC:>±  TIOJV. 
INDXJCTION^  ^I^r)   DEB^INITIONS. 

54*  1.  James  found  4  hens'  nests  in  the  bam,  and  in  each  nest 
were  5  eggs.     How  many  eggs  did  he  find  ? 

2.  If  a  cooper  can  make  7  barrels  in  a  day,  how  many  barrels  can 
lie  make  in  5  days  ? 

3.  How  many  blades  in  9  4-bladed  knives  ? 

4.  A  lady  bought  7  spools  of  thread,  at  7  cents  a  spool.  How 
much  did  it  cost  her  ? 

5.  If  9  pounds  of  flour  will  last  a  family  one  week,  how  many 
pounds  will  last  them  5  weeks  ? 

6.  How  many  dollars  can  a  man  earn  in  6  days,  if  he  earns  $3  a 
day? 

55.  J^fulHpticaHon  is  a  short  process  of  finding  the 
sum  of  as  many  times  one  of  two  numbers  as  there  are  ones 
in  the  other. 

56.  The  ^rodtici  is  the  result  obtained  by  Multiplica- 
tion. 

57.  The  jF^act07*S  are  the  numbers  used  to  obtain  the 
product. 

58.  The  J)€ictHplicand  is  that  factor  which  is  to  be 
taken  any  certain  number  of  times. 

59.  The  Muttiptier  is  that  factor  which  shows  how 
many  times  the  multiplicand  is  to  be  taken. 

60.  Continued  Mu2tip2icaHo7i  is  the  process  of  find- 
ing the  product  of  more  than  two  factors. 

61.  The  Sig7i  of  MulHpticaHo7Z^im.diQ  thus  x,  when 
placed  between  two  numbers,  signifies  that  they  are  to  be 
multiplied  together.  It  is  read  "times,"  or  "multiplied 
by."  Thus,  5  x  8  is  read  "5  times  8,"  or  "5  multiplied 
by  8." 


36 


INTEGERS. 


7.  4  X  6  slates  are  how  many  slates  ? 

8.  What  is  the  product  of  6  x  7  oranges  ? 

9.  What  is  the  product  of  7  x  9  ? 

10.  The  factors  are  5  and  8.     What  is  the  product  ? 

11.  The  multiplicand  is  9,  and  the  multiplier  8.     What  is  the 
product  ? 

13.  What  is  the  product  of  3  x  3  x  7  ? 

13.  4  X  2  X  6  pen-holders  =  how  many  pen-holders  ? 

62.      MULTIPLICATION    TABLE. 


0    1 

23456789 
11111111 
23456    7    89 

oiO    123456789 

'^|6    666666666 

0    6  12  18  24  30  36  42  48  54 

nil 

0    2 

23456789 

32222222 
4    6    8  10  12  14  16  18 

«(0123456    789 

*]7    777777777 
0    7  14  21  28  35  42  49  56  63 

qjO      1 

'^(3    3 
0    3 

23456789 
33333333 
6    9  12  15  18  21  24  27 

q(0123456789 

0]8    888888888 

0    8  16  24  32  40  48  56  64  72 

0    4 

23456789 
444444    44 
8  12  16  20  24  28  32  86 

q(01234    5    6789 

^\9    999999999 

0    9  18  27  36  45  54  63  72  81 

KJO    123456789 
^15555.5    55555 

0    5  10  15  20  25  30  35  40  45 

in^  0123456789 

•^"ho  10  10  10  10  10  10  10  10  10 

0  10  20  SO  40  50  60  70  SO  90 

CA.SE     I. 

The  Multiplier  One  Figure. 

673  +  673  +  673  +  673,  or  4 


63*  Ex.  How  many  are 
times  673  ? 

Explanation. — ^In  the  Solution  by  Addition  we 
find  the  sum  of  4  673's  to  be  2,692.  But  since 
3  ones  +  3  ones  +  3  ones  +  3  ones  =  4  times 
3  ones,  and  7  tens  -f-  7  tens  -f  7  tens  H-  7  tens 
=  4  times  7  tens,  and  6  hundreds  -}-  6  hundreds 
+  6  hundreds  4-  6  hundreds  =  4  times  6  hun- 


80LTTTI0N 
BY  AUDITION. 

673 
673 
673 
673 

2692 


MULTIPLICATION.  37 

dreds,  we  write  the  673  but  once,  and  write  a  4  under  its 
right-hand  figure,  to  show  how  many  times  it  is  to  be 
taken,  as  shown  in  the  following  Solution  by  Multipli- 
cation : 

^n  this   Solution  we  multiply  each    ^^  ^['^^Jl^^,,,,^, 
figure  of  the  multipHcand  by  the  multi-         ^  7  ^  Multiplicand. 
plier.      Thus,  4  times  3  ones  are  12  Jf.  ituiupUer. 

ones,  or  2  ones  and  1  ten.  We  write  26  92  Product. 
the  2  ones  as  the  ones  of  the  product, 
and  reserve  the  1  ten  in  the  mind  to  be  added  to  the  prod- 
uct of  the  tens.  4  times  7  tens  are  28  tens,  and  28  tens  + 
1  ten  =  29  tens,  or  9  tens  and  2  hundreds.  We  write  the 
9  tens  as  the  tens  of  the  product,  and  reserve  the  2  hun- 
dreds in  the  mind  to  be  added  to  the  product  of  the  hun- 
dreds. 4  times  6  hundreds  are  24  hundreds,  and  24  hun- 
dreds +  2  hundreds  =  26  hundreds,  or  6  hundreds  and  2 
thousands.  We  write  the  6  and  2  as  the  hundreds  and 
thousands  of  the  product.  The  result,  2,692,  is  the  sum  or 
product  required. 

64.  From  this  explanation  we  learn  that 

The  muUiplicand  is  that  factor  which  icould  he  used  in 
solving  a  problem  or  example  by  Addition. 

65.  We  may  add  abstract  numbers  or  like  concrete  num- 
bers.    (See  40.)     Hence, 

The  muUiplicand  may  be  either  an  absti^act  or  a  concrete 
number. 

66.  The  multiplier  is  used  simply  to  show  how  many 
limes  the  multiplicand  is  taken.     Hence, 

The  multiplier  is  alivays  an  abstract  number. 

67.  In  Addition  the  sum  is  of  the  same  kind  as  the  parts 
added.     Hence, 

The  product  is  always  of  the  same  kind  as  the  multiplicand. 


38  INTEGERS. 
FROBJOEMS. 

(1)        (2)          (3)          (4)            (5) 
Multiply         43        133        491        6243        13562  pounds, 
by         A        ^1        -1        1        ? 

6.  How  much  "will  3  cows  cost,  at  $52  apiece  ? 

7.  How  much  will  a  mechanic  earn  in  6  months,  if  he  earns  $35 
a  month  ?  $210. 

8.  A  grocer  bought  7  barrels  of  sugar,  each  containing  245 
pounds.     How  much  sugar  did  he  buy  ?  1, 715  pounds. 

9.  How  much  will  9  tons  of  hay  cost,  at  $14  a  ton  ?  $126. 

10.  How  far  will  a  railroad  train  run  in  7  hours,  at  the  rate  of 

39  miles  an  hour  ?  273  miles. 

11.  A  canal-boat  captain  bought  4  horses,  paying  $156  apiece  for 
them.     How  much  did  they  cost  him  ?  $62Ji,. 

12.  A  manufacturer  pays  his  hands  $2,356  a  month.  How  much 
do  their  wages  amount  to  in  6  months  ?  $H,  136. 

13.  In  a  certain  army  corps  there  were  8  regiments  of  966  men 
each.     What  was  the  number  of  men  in  the  corps  ?  7,728. 

14.  At  $5  a  week,  how  much  will  a  year's  board  cost,  there  being 
52  weeks  in  a  year  ?  $260. 

15.  If  the  price  of  thrashing  wheat  is  4  cents  a  bushel,  how  much 
must  be  paid  for  thrashing  17,944  bushels  ?  71,776  cents. 

16.  A  miller  sells  74  barrels  of  flour,  at  $9  a  barrel.  How  much 
does  the  sale  amount  to  ?  $666. 

68.  In  explaining  the  solution  of  Problem  16,  we  would 
say,  "  74  barrels  sell  for  74  times  as  much  as  1  barrel,  or  74 
times  $9."  Hence,  $9,  being  a  concrete  number,  is  the  true 
multiplicand.  But  since  74  times  9  is  the  same  as  9  times 
74,  in  solving  the  problem  we  may  take  74  for  the  multipli- 
cand, and  use  9  as  the  multiplier.     That  is, 

I.  In  the  solution  of  problems,  either  factor  may  he  used  as 

the  multiplicand.        See  Manual. 

n.  In  the  explanation  of  the  solution  of  problems  containing 
concrete  number's,  the  concrete  number  is  the  true  multiplicand. 


MULTIPLICATION. 


39 


17.  How  many  windows  in  the  front  of  all  the  stories  of  this 
building  except  the  first,  there  being  20  windows  to  each  story? 

18.  How  many  iron  pillars  or  columns  in  the  front  of  the  three 
upper  stories,  there  being  21  columns  to  each  story  ? 

19.  How  many  windows  in  3  street  cars,  there  being  18  windows 
in  each  car  ? 

20.  If  24  passengers  ride  in  a  car  at  one  trip,  and  the  fare  is  6 
cents,  how  much  fare  will  the  conductor  collect  ? 

21.  A  woman  who  keeps  a  fruit  stand  sold  178  oranges  one  day, 
at  4  cents  apiece.    How  much  did  she  receive  for  them  ? 

22.  How  much  did  it  cost  to  paint  the  two  signs  on  this  build- 
ing, at  $3  a  letter  ? 

23.  How  much  would  it  cost  all  the  persons  in  the  foreground 
of  this  picture  to  go  to  Harlem  and  back  on  a  horse-car,  the  fare 
being  7  cents  each  way  ?  210  cents. 


40  INTEGERS. 

24.  The  Third  Avenue  Railroad,  in  New  York  City,  is  7  miles 
long.     How  many  miles  does  a  car  run  in  making  11  round  trips  ? 

25.  A  wood  dealer  sold  6,591  cords  of  wood,  at  $7  a  cord.    How 
much  were  his  receipts  ?  $^6, 137. 

26.  In  1  mile  there  are  1,760  yards,  or  5,280  feet.     How  many 
yards  in  8  miles  ?  14,080. 

27.  How  many  feet  in  8  miles  ?  42,246. 

28.  Ten  brick-layers  finish  the  walls  of  a  building  in  9  days,  lay- 
ing 9,475  bricks  each  day.     How  many  bricks  are  in  the  walls  ? 

29.  At  $9  apiece,  how  much  will  it  cost  for  the  transportation 
of  33,875  soldiers  from  Washington  to  Chicago  ?  $304,875. 


C^SE     II. 

The   Multiplier   any  number   of  tens,   hundreds,  thousands, 
and  so  on. 

69,  Ex.  Multiply  3,176  by  10. 

Explanation. — ^We  write  the  numbers  as  shown  bolution. 

in  the  Solution,  and  multiply  each  figure  of  the  3176 

multiplicand  by  the   multiplier,  as   in   Case  I.  ^^ 

Comparing  the   multiplicand   and  product,  we  3176  0 
find  that  the   figures  of  ihe  product   are  the 
same  as  those  of  the  multiplicand,  with  a  cipher  on  the 
right.     Hence, 

70.  Annexing  a  cipher  to  any  number  multiplies  it  hy  10. 

71.  Annexing  a  second  cipher  multiplies  by  10  again  ;  that 
is,  annexing  two  ciphers  to  any  number  multiplies  it  by  10  times 
10,  or  100. 

72.  Annexing  three  ciphers  to  a  number  multiplies  it  by  10 
times  100,  or  1,000. 

73.  Annexing  four  ciphers  multiplies  by  10,000 ;  annex- 
ing Jive  ciphers,  by  100, 000 ;  and  annexing  six  ciphers,  by 
1,000,000, 


I 


MULTIPLICATION.  41 

74.  Ex.  Multiply  291  by  60. 
Explanation. — 60,  or  6  tens,  =  6  times  10  or        soltttion. 
10  times  6.     Hence,  60  times  291  is  10  times  as  291 

much  as  6  times  291.     We  may  therefore  multi-  ^^ 

ply  291  by  6,  and  to  the  product  thus  obtained       17  Jf6  0 
annex  a  cipher. 

75.  To  multiply  hy  600,  we  multiply  hy  6,  and  annex  two 
ciphers  to  the  product ;  to  multiply  hy  6,000,  we  multiply  hy  6, 
and  annex  three  ciphers. 

76.  We  multiply  hy  any  number  of  tens,  hundreds^  thousands, 
or  units  of  higher  orders  in  the  same  manner. 

Pit  O  BJjEMS. 

30.  How  many  oats  in  10  wagon  loads  of  65  bushels  each  ? 

31.  A  barrel  of  flour  contains  196  pounds.  How  many  pounds 
in  100  barrels  ?  19,600. 

82.  The  United  States  Government  purchased  10,000  rifles,  at 
$24  each.     How  much  did  they  cost  ?  $U0, 000. 

33.  Multiply  23,947  by  1,000.  Product,  23,947,000. 

34.  Multiply  175,941  by  100,000.  Product,  17,594,100,000. 

35.  A  pork  packer  sells  to  a  provision  dealer  125  barrels  of  pork, 
at  $20  a  barrel.    What  is  the  amount  of  his  bill  ?  $2,500. 

36.  At  $50  an  acre,  how  much  will  a  farm  of  176  acres  cost  ? 

37.  A  gentleman  bought  a  city  lot,  having  a  front  of  22  feet,  at 
$90  a  foot.     How  much  did  it  cost  him  ?  $1,980. 

38.  A  drover  sold  500  head  of  cattle,  at  $67  a  head.  How  much 
did  he  receive  ?  $33, 500. 

39.  A  manufacturer  sells  594  sewing-machines,  at  $60  each. 
How  much  does  he  receive  for  them  ?  $35, 640. 

40.  In  a  certain  county,  237  drafted  men  purchased  their  exemp- 
tion by  paying  $300  each.     How  much  did  their  exemption  cost  ? 

41.  The  President's  Cabinet  consists  of  7  members,  who  receive  a 
salary  of  $8,000  each.     "What  do  their  united  salaries  amount  to  ? 

42.  400  X  495  =  how  many?  198,000, 

43.  What  is  the  product  of  32,721  x  50,000  ?     1,636,050,000. 


42 


INTEGERS, 


C-A.sk     III 


The  Multiplier  more  than  One  Figure. 
77.  Ex.  Multiply  3,528  by  472. 


Multiplying 

S528 


7056 


Explanation.— Since 
472  consists  of  2  ones, 
7  tens,  and  4  hun- 
dreds, or  2,  70,  and 
400 ;  and  since  we  can 
not  multiply  3,528  by 
the  whole  472  at  once,  we  multiply 
it  first  by  2,  then  by  70,  and  then 
by  400,  and  afterward  add  the  re- 
sults, or  Partial  Products.  The  re- 
sult thus  obtained,  1,665,216,  is  the 
sum  of  2  X  3,528,  70  x  3,528,  and 
400  X  3,528,  or  472  x  3,528. 

In  the  First  Solution,  each  of  the 
four  steps  stands  by  itself  ;  in  the 
Second  Solution  they  are  placed  to- 
gether. 

In  the  Second  Solution,  the  ciphers 
on  the  right  of  the  second  and  third 
partial  products  serve  merely  to  fill 
the  places  of  ones  and  tens  ;  and  since 
the  sum  of  any  number  of  O's  is  0, 
they  may  be  omitted  without  affect- 
ing the  total  product,  as  shown  in  the 
Third  Solution. 

In  this  Solution  the  second  partial 
product  is  found  by  multiplying  by 
7,  instead  of  70 ;  and  the  third  partial 
product  by  multiplying  by  4,  instead 
of  400.     But  we  must  always 


FIEST  SOLTTTION. 


Multiplying 
hy  TO. 

8528 

7^ 

2^6960 


Multiplying 

lyma. 
3528 
J^OO 

lJt.11200 


Adding 
Partial  Products. 

7056 

246960 

1^11200 

1665216 


SECOND   SOLUTION. 

3528 
Ji.72 


Partial 
Products, 


7056' 

246960 

1411200 

1665216 


TIIIED  SOLXmON. 

3528 
472 


Partial 
Products. 


7056 
24696 
14112 

1665216 


Partial 
Products. 


MULTIPLICATION.  43 

Write  the  first  figure  of  each  partial  product  directly  under 
the  figure  of  the  multiplier  used  to  obtain  it. 

Note.— In  the  explanation  of  Solutions,  the  value  of  each  partial  product 
should  be  named.  This  is  done  by  reading  it  as  though  the  ciphers  were 
written. 

PR  OBZJSMS. 

(44)  (45)  (46)  (47)  (48) 

34  73  281  2976  127493 

23  45  _54  ^  647 

49.  A  prairie  farmer  planted  176  acres  of  com,  wliich  yielded  73 
bushels  to  the  acre.     How  many  bushels  in  the  crop  ?       12,8J^8. 

50.  At  a  certain  recruiting  station,  in  1862,  43  men  enlisted  each 
day  for  36  days.     How  many  men  enlisted?  l,5JiS. 

51.  An  overland  emigrant  train  traveled  23  miles  a  day  for  17 
days.    How  far  did  they  travel  ?  S91  miles. 

52.  The  skipper  of  a  fishing  smack  received,  as  his  share  of  the 
season's  catch,  273  barrels  of  mackerel,  which  he  sold  at  $11  a  bar- 
rel.   How  much  were  his  receipts  ?  $3, 003. 

53.  In  a  pump  manufactory,  125  of  the  workmen  receive  $39  a 
month  each.     How  much  do  their  wages  for  a  month  amount  to  ? 

54.  How  much  do  their  wages  amount  to  in  1  year,  or  12 
months?  $58,500. 

55.  A  livery-stable  keeper  who  has  23  horses,  finds  that  it  costs 
him  $83  a  year  for  the  keeping  of  each  horse.  What  is  the  cost  of 
keeping  all  of  them  ?  $1,909. 

56.  A  steamer  sailed  from  New  York  for  Liverpool  with  376 
first-class  passengers.  How  much  did  their  fares  amount  to,  at 
$135  apiece?  $50,760. 

57.  A  railroad  company  contracted  for  43  locomotives,  at  $18,725 
apiece.    What  was  the  amount  of  the  contract  ?  $805, 175. 

58.  In  a  certain  paper-mill  184  reams  of  paper  are  made  daily. 
How  many  reams  are  made  in  a  year  of  313  working  days  ? 

59.  Multiply  1,372  by  861.  Product,  1,181,292. 

60.  What  is  the  product  of  4,293  multiplied  by  2,726  ? 

11,702,718. 

61.  417,293  X  581  =  how  many?  242,U7,233. 


44  ,  INTEGERS. 

One  or  more  ciphers  bet-ween  other  figures  of  the  Multiplier. 
78.  Ex.  Multiply  2,566  by  3,007. 
Explanation. — In  the  Second     ^'^'^"^  solution,     second  solxttion. 
Solution  we  have  multiplied  by  2566  2566 

7  ones  and  3  thousands   (see  ^^^^  JOOJ 

76),  omitting  to  multiply  byO         17962  17962 

X  ^A-u        ^       ^      X  n  0000  7698 

tens  and  0  hundreds,  because  0       OOOO  

times  2,566  is  0,  as  shown  in  '^^gg                  7  7 159  62 

the  First  Solution.     Hence  we  ^  ^^^^ 
may  always 

Omit  to  multiply  by  ciphers  that  stand  between  other  figures 
in  the  multiplier. 

PROBLEMS. 

(62)  (63)  (64)  (65) 

Multiply    426  1728  4765  29873 

by    203  _506  _807  5008 

66.  At  $105  each  for  horses,  how  much  will  it  cost  to  moitnt  a 
cavalry  regiment  of  1,043  men  ?  $109,515. 

67.  In  one  season  a  manufacturer  sold  307  reapers,  at  $135 
apiece.    How  much  did  he  receive  for  them  ?  $41,445- 

68.  A  furrier  bought  108  buffalo-robes,  at  $17  apiece?  How 
much  did  they  cost  him  ?  $1,836. 

69.  A  canal  203  miles  long  was  kept  in  repair  one  season  at  an 
expense  of  $383  a  mile.  What  was  the  expense  for  the  whole 
canal?  $77,749. 

70.  At  a  cotton  manufactory  1,396  yards  of  cloth  are  made  each 
day.     How  many  yards  are  made  in  307  days  ?  4^8j572. 

71.  In  a  certain  cotton  factory  are  203  looms,  which  turn  out  69 
yards  of  cloth  per  day,  each.  How  many  yards  of  goods  are  made 
daily  ? 

72.  At  the  same  rate,  what  is  the  total  product  of  the  factory  in 
a  year  of  308  working  days  ?  4j^Hj  ^^6  yards. 


M  ULTIPLIC  ATION, 


45 


One  or  more  ciphers  on  the  right  of  either  or  both  factora 


79.  Ex.  1.  Multiply  89  by  1,600. 
To  multiply  by  1,600,  we  first  multiply  by 
16,  and  then  to  the  product  annex  two  ciphers, 
as  in  Case  II.     (See  75.) 

Ex.  2.  Multiply  54,000  by  37. 

,  Explanation. — "We  first  multiply  54  by  37, 
and  obtain  1998.  Since  the  54  is  thousands, 
this  product  must  be  thousands  (see  67) ;  and 
we  therefore  write  three  ciphers  in  units'  pe- 
riod ;  that  is,  annex  three  ciphers  to  the  prod- 
uct of  37  X  54. 


SOLUTION. 

89 
1600 

89 
lJi.2Ji.00 


BOLUTIOIf. 

5Jf.000 

87 

378 
162    

1998000 


60LCTI0N. 

73000 
2600 

Jf.38 
lJf6 

189800000 


Ex.  3.  Multiply  73,000  by  2,600. 
Explanation. — We  first  multiply  73  by 
28,  obtaining  1,898.  But  since  the  73  is 
thousands,  we  annex  three  ciphers  to  this 
product  for  those  at  the  right  of  the  mul- 
tiplicand.    The  result   thus   obtained  is 

26  X  73,000 ;   and  to  make   it  2,600  x 
73,000,    we    annex    two     ciphers    more. 

Hence,  when  there  are  ciphers  on  the  right  of  one  or  both 
factors. 

To  the  product  of  the  other  figures  annex  as  many  ciphers  as 
there  are  ciphers  on  the  right  of  both  factors. 

PROBLEMS. 

73.  In  one  barrel  of  beef  there  are  200  pounds.     How  many 
pounds  in  37  barrels  ?  7jJi,00. 

74.  A  carriage  maker  sold   30  top  carriages,   at  $285   apiece. 
How  much  did  they  come  to  ?  $8,550. 

75.  A  Missouri  farmer  had  130  acres  of  wheat,  ajtd  the  yield  was 

27  bushels  to  the  acre.    How  mu^h  wheat  did  hef raise  ?  ^.^^  TH-^  '"''^  - 


46  INTEGERS. 

76.  In  printing  an  edition  of  5,000  copies  of  a  book,  16  sheets  of 
paper  were  used  for  each  book.    How  much  paper  vf as  used  ? 

77.  A  Georgia  planter  had  94  acres  of  cotton  that  yielded  460 
pounds  to  the  acre.     How  much  cotton  did  he  raise  ? 

78.  At  an  ax  factory  280  axes  were  made  each  day  for  156  days. 
How  many  axes  were  made  ?  Jf3, 680. 

79.  What  will  be  the  cost  of  25,000  army  blankets,  at  $5  apiece  ? 

80.  In  one  ream  of  paper  there  are  480  sheets.  How  many  sheets 
in  260  reams  ?  124,800. 

81.  A  Pennsylvania  oil  well  flowed  110  days,  at  the  rate  of  340 
barrels  of  oil  a  day.     How  much  oil  did  it  yield  ?    37,400  larrels. 

82.  How  much  will  be  the  cost  of  building  a  line  of  telegraph 
680  miles  long,  at  $1,250  a  mile  ?  $850,000. 

83.  What  is  the  product  of  760,000  and  53,000  ?  40,280,000,000. 

84.  The  multiplicand  is  694,000,  and  the  multiplier  56,700. 
What  is  the  product  ?  39,349,800,000. 

80.  The  facts  deduced  in  Arts.  64  to  70  may  be  stated  as 

Principles  of  Mtcltiplication, 

In  the  Explanation  of  Solutions  : 

I.  The  true  multiplicand  is  that  factor  which  mould  be  used 
in  solving  the  problem  by  Addition. 

II.  The  multiplicand  may  be  either  an  abstract  or  a  concrete 
number. 

m.   The  multiplier  must  always  be  an  abstract  number. 

IV.  Tlie  product  is  always  of  the  same  kind  as  the  true  mul- 
tiplicand. 

In  tbe  Solution  of  Problems  : 

V.  In  the  solution  of  problemSy  either  factor  may  be  used  as 
the  multiplicand. 

VI.  Annexing  a  cipher  to  any  number  multiplies  it  by  10. 

VII.  The  sum  of  all  the  partial  products  arising  from  multi- 
plying one  of  two  numbers  by  the  ones,  tens,  hundreds,  etc.,  of 
the  other,  is  the  product  of  the  two  given  numbers. 


MULTIPLICATION.  47 

81.  Upon  these  principles  is  based  the 

^ute  for  MuUipUcation  of  Integers. 

I.  The  multiplier  one  figure. 

Commence  with  the  ones,  and  multiply  successively  each  figure 
of  the  multiplicand  by  the  multiplier.  In  the  product  write  the 
ones  of  each  result,  and  add  the  tens  to  the  next  result. 

II.  The  multiplier  more  than  one  figure. 

Multiply  by  each  figure  of  the  multiplier  except  ciphers,  place 
the  right-hand  figure  of  each  product  under  that  figure  of  the 
multiplier  used  to  obtain  it,  and  add  the  partial  products. 

m.  Ciphers  on  the  right  of  either  or  both  factors. 

To  the  product  of  all  the  other  figures  annex  as  many  ciphers 
as  there  are  ciphers  on  the  right  of  both  factors. 

FR  OB  IjEMS. 

85.  Every  mile  of  a  road  4  rods  wide  contains  8  acres.  How- 
many  acres  are  there  in  168  miles  of  such  a  road  ?  1^344' 

86.  An  army  train  was  9  hours  in  passing  a  given  point,  84 
wagons  passing  each  hour.     How  many  wagons  were  in  the  train  ? 

87.  How  many  poles  will  be  required  for  a  telegraph  328  miles 
long,  if  16  poles  are  required  for  one  mile  ?  5,^48. 

88.  A  phonographic  reporter,  in  taking  down  a  speech,  wrote  68 
words  a  minute  for  137  minutes.  How  many  words  were  in  the 
speech?  9,316. 

89.  How  much  will  it  cost  to  build  a  railroad  134  miles  long,  at 
$65,475  a  mile  ?  $8, 773, 650. 

90.  Last  season  a  cheese  factory  used  the  milk  of  470  cows,  and 
made  290  pounds  of  cheese  to  the  cow.  How  much  cheese  was 
made  ?  136, 300  pounds. 

91.  In  one  square  mile  there  are  640  acres,  and  in  the  State  of 
Iowa  there  are  50,914  square  miles.  How  many  acres  in  the 
State?  32,584,960. 

92.  The  New  American  Cyclopsedia  contains  13,805  pages,  of 
3,036  ems  each.    How  many  ems  in  the  work  ?  41,911,980. 

93.  The  factors  are  three  hundred  ninety-seven  thousand  five 
hundred,  and  nine  thousand  eight  hundred.  What  is  the  product  ? 
Three  Mllion  eight  hundred  ninety-five  million  five  hundred  thousand. 


48  INTEGERS. 

94.  How  many  miles  will  a  railroad  conductor  travel  in  a  year, 
if  he  goes  over  a  road  108  miles  long  once  every  day  ?       39,  ^20. 

95.  It  is  estimated  that  Mississippi  River  deposits  3,703,758,400 
cubic  feet  of  solid  matter  in  the  Gulf  of  Mexico  evei*y  year.  How 
many  cubic  feet  have  been  deposited  in  503  years  ? 

1,858,784,716,800. 

96.  How  many  yards  of  sheeting  are  there  in  91  bales,  each  bale 
containing  33  pieces  of  45  yards  each  ?  94j  185. 

97.  At  $13  a  month,  how  much  will  the  pay  of  the  privates  of  a 
certain  regiment,  896  in  number,  amount  to  in  13  months  ?  $139,776. 

98.  A  shoe  dealer  bought  37  cases  of  French  calf  boots,  each  case 
containing  13  pairs,  at  |5  a  pair.  What  was  the  amount  of  the 
purchase?  $1,620. 

99.  In  a  woolen  factory  there  are  48  looms.  How  many  yards  of 
cloth  will  be  made  in  the  factory  in  308  days,  if  37  yards  are  woven 
upon  each  loom  daily?  269,568. 

100.  Just  before  an  expected  battle,  04  rounds  of  cartridges  were 
given  to  each  of  70,000  men.  How  many  rounds  were  distributed 
to  all  of  them  ?  4, 480, 000. 

Po-wers. 

82.  A  ^ower  is  the  product  of  two  or  more  equal  fac- 
tors ;  as  49,  which  is  the  product  of  7  x  7. 

83.  A  Square  is  the  product  of  two  equal  factors  ;  as 
25,  which  equals  5x5. 

84.  A  Cube  is  the  product  of  three  equal  factors ;  as  64, 
which  equals  4x4x4. 

85.  The  JF'ourtli  ^ower\^  the  product  of  four  equal  fac- 
tors ;  the  ^iftJi  ^ower  the  product  of  five  equal  factors  ; 
the  Sixt?i  ^07Per  the  product  of  six  equal  factors,  and 
so  on. 

Notes.— 1.  The  process  of  finding  the  square  of  a  number  is  called  squar- 
ing it ;  and  the  process  of  finding  its  cube  is  called  cubing  it. 

3.  The  process  of  finding  any  power  of  a  number  is  called  raising  it  to 
that  power. 


MULTIPLICATION.  49 

86.  An  Index  is  that  one  of  two  numbers  which  denotes 
the  power  to  which  the  other  number  is  to  be  raised.  It  is 
written  at  the  right,  and  a  httle  above  the  other  number. 
Thus,  in  the  expression  8^  3  is  an  index,  and  it  denotes 
that  8  is  to  be  cubed.  So,  also,  IV  indicates  the  square  of 
21,  and  59*  indicates  the  fourth  power  of  59. 

87.  Ex.  Kaise  15  to  the  fourth  power.       solution. 
Explanation.  —  We   first  find  the  ^  ^ 

square  of   15,  by  multiplying  it  by  _ 

itself.     We  then  find  the   cube  by  7^ 

multiplying  the  square  by  15.     We  -^^ 

then  multiply  the  cube  by  15,  and  the  22  5  Square. 

result  is  the  fourth  power ;  because  -^^ 

it  is  the  product  of  15  x  15  x  15  x  15.  1125 

In   squaring  a  number  there  are  ^^^ 

2   equal  factors   and    1    multipHca-  3  3  7  5  Cube. 

tion  ;   in  cubing  it,  3   equal  factors  ^  ^ 

and  2  multiplications  ;  and  in  raising  16  8  7  5 

it  to  the  fourth  power,  4  equal  fac-  3  375 

tors  and  3  multiplications.  5  0  6  2  5  FourVipower. 

In  finding  any  power  of  a  number ^  the  number  of  multiplica- 
tions is  one  less  than  the  number  of  factors, 

PB,  OBLJEMS, 

101.  What  is  the  square  of  9  ?  SI. 

102.  What  is  the  cube  of  5  ?  125. 

103.  Square  423.     Cube  47.  178,929;  103,823. 

104.  Raise  12  to  the  fourth  power.  20,736. 

105.  What  is  the  cube  of  52  ?    The  cube  of  901  ? 

106.  What  is  the  square  of  2,016  ?  4^064,256. 

107.  Raise  218  to  the  sixth  power.  107,334,407,093,824. 

108.  Raise  63  to  the  seventh  power. 

109.  Raise  14  to  the  eighth  power. 

110.  Raise  the  following  numbers  to  the  powers  denoted  by  their 
.  18S  11^ 

3 


50  INTEGERS. 

SECTION    VI. 

^ITISIOJV, 
iNDXJCTionsr  ^nd  definitions. 

88.  1.  Emma  exchanged  a  50-cent  fractional-currency  note  for 
5-cent  pieces.     How  many  5-cent  pieces  did  she  receive  ? 

3.  How  many  oranges  can  I  buy  for  28  cents,  if  I  pay  4  cents 
apiece  for  them  ? 

3.  A  silversmith  sold  30  teaspoons,  in  sets  of  6  spoons  each. 
How  many  sets  of  spoons  did  he  sell  ? 

4.  A  farmer  put  15  bushels  of  oats  into  bags,  putting  3  bushels 
in  each  bag.     How  many  bags  did  he  use  ? 

5.  One  day  Henry  saw  a  pic-nic  party  of  36  persons  passing  by, 
in  4  carriages.     How  many  persons  were  there  for  each  carriage  ? 

6.  Two  boys  received  20  cents  for  carrying  a  lady's  trunk  to  the 
depot,  and  they  shared  the  money  equally  between  them.  How 
many  cents  had  each  boy  ? 

7.  In  my  garden  are  21  fruit-trees,  standing  in  3  equal  rows. 
How  many  trees  in  1  row  ? 

8.  A  grocer  paid  $54  for  9  barrels  of  cranberries.  What  was  the 
price  per  barrel  ? 

In  solving  each  of  the  first  four  problems,  we  find  how 
many  times  one  of  two  numbers  is  contained  in  the  other  ; 
and  in  solving  each  of  the  other  problems,  we  separate  one 
of  two  numbers  into  as  many  equal  parts  as  there  are  ones 
in  the  other. 

89.  !Dirtsio?z  is  the  process  of  finding  how  many  times 
one  of  two  numbers  is  contained  in  the  other  ;  or  of  finding 
one  of  the  equal  parts  into  which  a  number  may  be  divided. 

90.  The  Quotient  is  the  result  obtained  by  Division. 

91.  The  dividend  is  the  number  to  be  divided. 

92.  The  divisor  is  the  number  by  which  the  dividend 
is  to  be  divided. 


DIVISION. 


51 


NoTES.—l.  A  Partial  Dividend  is  that  part  of  the  dividend  used  to  obtain 
one  figure  of  the  quotient,  when  the  whole  dividend  is  too  large  to  obtain 
the  entire  quotient  at  one  operation. 

2.  Division  is  Exact  when  all  the  dividend  is  divided  and  the  quotient  is 
a  whole  number. 

3.  A  Remainder  is  that  part  of  the  dividend  left  undivided,  either  when 
the  division  is  only  partiaUy  completed,  or  when  exact  division  is  impos- 
sible. 

93.  The  Sign  ofDivision,  made  thus  -^,  when  placed 
between  two  numbers,  signifies  that  the  number  before  it  is 
to  be  divided  by  the  number  after  it.  It  is  read,  "divided 
by."    Thus,  175  -~  25  is  read  "  175  divided  by  25." 

Notes. — 1.  Division  is  also  expressed  by  writing  the  dividend  above,  and 
the  divisor  below  a  horizontal  line.    Thus,  ^l§-  is  read,  "  175  divided  by  25." 

2.  In  writing  numbers  for  solution,  the  divisor  may  be  written  either  at 
the  right  of  the  dividend,  thus,  175  1,25,  or  at  the  left  of  it,  thus,  25  J  175. 

9.  How  many  times  are  6  cents  contained  in  54  cents  ? 

10.  What  is  the  quotient  of  40  divided  by  8  ? 

11.  56  -T-  7  =  how  many  ?    -^  =  how  many  ? 

12.  The  dividend  is  24,  and  the  divisor  is  4.  What  is  the 
quotient  ? 

13.  If  as  many  of  17  apples  be  divided  among  5  children  as  will 
give  them  whole  apples,  how  many  whole  apples  will  each  child 
have ;  and  how  many  apples  will  be  the  remainder  ? 

14.  Divide  42  figs  among  6  girls,  and  tell  me  the  dividend,  the 
divisor,  and  the  quotient.     Why  is  there  no  remainder  ? 

94.     DIVISION    TABLE. 


0123456789  [1 
0123456789 

0  6  12  18  24  30  36  43  48  .54  [  6 
0123456789 

0    2    4    6    8  10  12  14  16  18  (^  2 
0123456789 

0  7  14  21  28  35  42  49  56  63  1  7 
0123456789 

0    3    6    9  12  15  18  21  24  27  1  3 
0123456789 

0  8  16  24  32  40  48  56  64  72  [  8 
0i23456789 

0    4    8  12  16  20  24  28  32  36  [4 
0123456789 

0  9  18  27  36  45  54  63  72  81  [ 9 
0123456789 

0    5  10  15  20  25  30  35  40  45  [ 5 
0123456789 

0  10  20  30  40  50  60  70  80  90  [IQ 
0123456789 

52 


INTEGERS. 


95.  2  is  contained  in  6,  3  times  ;  in  6  tens  or  60,  3  tens  or 
30  times;  in  6  hundreds  or  600,  3  hundreds  or  300  times.  25 
is  contained  in  75,  3  times  ;  in  75  tens  or  750,  3  tens  or  30 
times  ;  in  75  hundreds  or  7,500,  3  hundreds  or  300  times. 

In  other  words,  if  we  divide  ones,  the  quotient  must  be 
ones  ;  if  we  divide  tens,  the  quotient  must  be  tens  ;  if  we 
divide  hundreds,  the  quotient  must  be  hundreds  ;  if  we 
divide  thousands,  the  quotient  must  be  thousands,  and  so 
on.    Hence,  in  division  of  integers. 

Any  quotient  figure  must  he  of  the  same  name  or  order  of 
units  as  the  right-hand  figure  of  the  partial  dividend  used  to 

obtain  it.       see  ManuaL 


C^SE    I. 
The  Divisor  One  Figure. 


FIEST   METHOD. 


96.  Ex.  1.  Divide  936  by  3. 


SOLUTION, 

936 

0 


3   Divisor. 
312       Quotient. 


3 


Explanation.  —  We  place 
the  divisor  at  the  right  of  the 
dividend,  separating  them  by  Dividend. 
a  hne,  and  draw  a  hne  under 
the  divisor,  to  separate  it 
from  the  quotient.  Then 
commencing  at  the  left  hand,  ^ 

we  divide  each  figure  of  the  — 

dividend     by    the     divisor, 

thus :  3  is  contained  in  9,  3  times ;  and  as  the  9  is  hun- 
dreds, the  3  is  hundreds  (95),  and  we  write  it  as  the  first 
figure  of  the  quotient.  We  have  now  used  the  9  hundreds 
of  the  dividend,  and  subtracting  it  from  the  dividend,  we 
have  no  hundreds  left.  The  next  part  of  the  dividend  to  be 
used  is  the  3  tens,  which  we  bring  down  for  a  partial 
dividend.  3  is  contained  in  3,  1  time  ;  and  as  the  3  is 
tens,  the  1  is  a  ten  (95),  and  we  write  it  as  the  second  figure 


DIVISION.  53 

of  the  quotient.  We  have  now  used  the  3  tens  of  the  divi- 
dend, and  subtracting  it  from  the  dividend,  we  have  no 
tens  left.  The  next  part  of  the  dividend  to  be  used  is  the  6 
ones,  which  we  bring  down  for  another  partial  dividend. 
3  is  contained  in  6,  2  times ;  and  as  the  6  is  ones,  the  2  is 
ones  (95),  and  we  write  it  as  the  third  figure  of  the  quo- 
tient. We  have  now  used  all  the  figures  of  the  dividend, 
and  the  result,  312,  is  the  quotient  required. 

Ex.  2.  Divide  17,668  by  7. 
Explanation. — 7  is  not  contained  in  1 
any  number  of  times  ;  we  must  there- 
fore take  17  for  the  first  partial  dividend. 
Since  2  times  7  are  14  and  3  times  7  are 
21,  and  17  is  more  than  14,  but  less  than 
21,  7  is  contained  in  17,  2  times.  As  17 
is  thousands,  the  2  is  thousands  (95), 
and  we  write  it  as  the  first  or  thousands' 
figure  of  the  quotient.  We  have  now 
used  2  (thousands)  times  7,  or  14  (thous- 
ands) of  the  dividend ;  and  to  find  how  many  thousands 
remain  undivided,  we  subtract  the  14  (thousands)  from  the 
17  (thousands),  and  obtain  a  remainder  of  3  (thousands). 
We  next  bring  down  the  6  (hundreds)  of  the  dividend,  and 
uniting  it  with  the  3  (thousands),  we  have  36  (hundreds)  for 
a  second  partial  dividend.  Since  5  times  7  are  35,  and  6 
times  7  are  42,  and  36  is  more  than  5x7  and  less  than 
6  X  7,  7  is  contained  in  36,  5  times.  As  36  is  hundreds, 
the  5  is  hundreds  (95),  and  we  write  it  as  the  second  or 
hundreds'  figure  of  the  quotient.  We  have  now  used  5 
(hundreds)  times  7,  or  35  (hundreds)  of  the  partial  divi- 
dend ;  and  to  find  how  many  hundreds  remain  undivided, 
we  subtract  the  35  (hundreds)  from  the  36  (hundreds),  and 
obtain  a  remainder  of  1  (hundred).  We  next  bring  down 
the  6  (tens)  of  the  dividend,  and  uniting  it  with  the  1  (hun- 
dred),  we  have   16    (tens)   for  another  partial   dividend. 


SOLUTION 

17668 

7 

U 

252A 

86 

85 

16 

U 

28 

28 

54  INTEGERS. 

Since  16  is  more  than  2  times  7  and  less  than  3  times  7,  7 
is  contained  in  16,  2  times.  As  16  is  tens,  the  2  is  tens 
(95),  and  we  write  it  as  the  third  or  tens'  figure  of  the  quo- 
tient. We  have  now  used  2  (tens)  times  7,  or  14  (tens)  of 
the  last  partial  dividend ;  and  to  find  how  many  tens  re- 
main undivided,  we  subtract  the  14  (tens)  from  the  16 
(tens),  and  obtain  a  remainder  of  2  (tens).  We  now  bring- 
down the  8  (ones)  of  the  dividend,  and  uniting  it  with  the  2 
(tens),  we  have  28  for  a  final  partial  dividend.  7  is  con- 
tained in  28,  4  times  ;  and  as  28  is  ones,  the  4  is  ones  (95), 
and  we  write  it  as  the  last  or  ones'  figure  of  the  quotient 
We  have  now  used  4  times  7,  or  28  ;  and  subtracting  this 
from  the  last  partial  dividend,  we  have  no  remainder.  All 
the  figures  of  the  dividend  have  been  used  ;  and  the  result, 
2,524,  is  the  quotient  required. 

97.  The  quotient  in  Division  is  sometimes  an  abstract, 
and  sometimes  a  concrete  number.  It  is  therefore  neces- 
sary, before  proceeding  to  the  solution  of  problems,  that  we 
determine  when  the  quotient  is  abstract,  and  when  con- 
crete.    To  do  this,  we  will  take  the  following  examples  : 


(1) 

15  cents  [  3  cents 
5 

(2) 
-    1513 

5 

(3) 
15  cents  ^3 

5  cents 

(4) 
15  13  cents 

Impossible 

Writing  in  the  places  of  numbers,  words  indicating  the 
kinds  of  numbers  used,  we  have  : 

(1)  (2) 

Concrete  [  Cmicrete  Abstract  [  Abstract 

Abstract  Abstract 

(3)  (4) 

Cmicrete  yAbsPract  Abstract  [  Concrete 

Concrete  Impossible 


DIVISION.  55 

98.   These  illustrations  fully  establish  the  following  facts : 

I.  The  quotient  will  he  an  abstract  number,  when  the  divi- 
dend and  divisor  are  both  abstract  or  both  concrete  numbers. 
(Ex.  1,  2.) 

II.  The  quotient  will  be  a  concrete  number,  ivhen  the  dividend 
is  a  concrete,  and  the  divisor  an  abstract  number.     (Ex.  3.) 

III.  Either  the  divisor  or  the  quotient  must  always  be  an 
abstract  number.     (Ex.  1,  2,  3.) 

IV.  An  abstract  number  can  not  be  divided  by  a  concrete 
number.     (Ex.  4.) 

PBOBZEMS, 

(1)  (3)  (3)  (4) 

648  1 2_  8484  [$4  373  tons  t;^  7393  [6 

5.  A  stage  company  paid  $396  for  3  horses.  How  much  did 
they  cost  apiece  ?  $132. 

6.  How  many  suits  of  clothes  can  be  made  from  1,348  yards  of 
broadcloth,  allowing  4  yards  for  each  suit  ?  312. 

7.  At  a  mortgage  sale,  3  city  lots  were  sold  for  $1,596.  How 
much  was  that  for  one  lot  ? 

8.  A  railroad  company  bought  1,456  cords  of  wood,  which  they 
transported  on  platform  cars,  each  carrying  8  cords.  How  many 
car  loads  were  there  ?  182. 

9.  An  Ohio  farmer  raised  a  crop  of  1,965  bushels  of  wheat,  which 
he  exchanged  with  a  miller  for  flour,  receiving  1  barrel  of  flour  for 
every  5  bushels  of  wheat.  How  much  flour  did  he  receive  for  his 
wheat  crop  ?  393  barrels. 

10.  If  6  masons  lay  15,894  bricks  in  a  day,  how  many  bricks  can 
1  mason  lay?  2,649. 

11.  If  the  yearly  expenses  of  a  family  of  7  persons  are  $3,065, 
what  are  the  expenses  of  1  person  ? 

13.  An  army-wagon  train,  9  miles  long,  contains  1,944  wagons. 
How  many  wagons  is  that  to  the  mile  ?  216. 


56  INTEGERS. 

SECOND   METHOD. 

99.  Ex.  Divide  25,216  by  8. 

Explanation. — We  write  the 
dividend  and  divisor  as  in  the  bolutiox. 

First  Method,  but  below  the     ^i^i<^ena.^5 2  1  6  [Sm^isor. 
dividend  we  draw  a  horizontal  315  2  Quotient. 

Hne,  under  which  to  write  the 

quotient.  8  is  contained  in  25  (thousands),  3  (thousands) 
times.  This  quotient  figure  we  write  directly  below  the 
last  figure  (5)  of  the  part  of  the  dividend  used  to  obtain  it. 
We  multiply  the  divisor  (8)  by  the  3  (thousands),  and  sub- 
tract the  product  from  the  25,  performing  both  computa- 
tions mentally.  We  now  mentally  unite  the  remainder  1 
(thousand)  with  the  next  figure  of  the  dividend,  2  (hun- 
dreds), and  divide  the  result,  12  (hundreds),  by  the  divisor. 
8  is  contained  in  12  (hundreds)  1  (hundred)  time.  We 
write  the  1  as  the  second  figure  of  the  quotient,  and  then 
multiply  8  by  it,  and  subtract  the  product  from  the  partial 
dividend,  12  (hundreds),  performing  the  computations  men- 
tally, as  before.  We  next  mentally  unite  the  remainder,  4 
(hundreds),  with  the  1  (ten)  of  the  dividend,  and  divide  the 
result,  41  (tens),  by  the  divisor.  8  is  contained  in  41  (tens) 
5  (tens)  times.  We  write  the  5  as  the  third  figure  of  the 
quotient,  and  then  multiply  8  by  it,  and  subtract  the  prod- 
uct from  the  partial  dividend,  41  (tens),  performing  the 
computations  mentally,  as  before.  We  mentally  unite  the 
remainder,  1  (ten),  with  the  6  (ones)  of  the  dividend,  and 
divide  the  result  (16)  by  the  divisor.  8  is  contained  in  16 
(ones)  2  (ones)  times,  and  we  write  2  as  the  fourth  figure 
of  the  quotient.  We  have  now  used  all  the  figures  of  the 
dividend ;  and  the  result,  3,152,  is  the  quotient  required. 

In  the  Second  Method,  the  same  computations  are  ]3er- 
f ormed  as  in  the  First  Method  ;  but  the  results  of  the  sub- 
tractions and  multiplications  are  not  written,  and  hence 
fewer  figures  are  used. 


DIVISION 


57 


100.  I^ong  division  is  that  method  of  dividing,  in 
which  all  the  products  and  partial  dividends  are  written. 

101.  Short  1)ivision  is  that  method  of  dividing,  in 
which  only  the  dividend,  divisor,  and  quotient  are  written. 

Seo  Manual. 
VJt  O  BIjJEMS. 

13.  A  city  corporation  paid  $9,376  for  3  steam  fire-engines. 
What  was  the  cost  of  each  ?  $J^,  688. 

14.  A  man  whose  wages  were  $3  a  day,  earned  $891  in  a  year. 
How  many  days  did  he  work  ?  297. 

15.  A  steamboat  which  was  owned  by  4  men  in  equal  shares,  was 
sold  for  $39,724.  How  much  was  each  man's  share  of  the  re- 
ceipts? $9,931. 

16.  A  farmer  has  1,458  bush- 
els of  wheat,  which  he  intends 
to  carry  to  market  in  2-bushel 
bags.  How  many  bagfuls  will 
he  have?  729. 

17.  If  3  horses  eat  2,025 
pounds  of  hay  in  a  month,  how 
much  will  1  horse  eat  ? 

18.  If  I  feed  a  horse  7  half- 
bushel  measures  of  oats  in  a 
week,  how  many  weeks  will 
483  half-bushels  last  him  ? 

19.  How  long  will  3  horses 
be  in  eating  the  same  quantity 
of  oats  ?  23  weeks. 

20.  In  a  barrel  in  the  gran- 
ary are  96  quarts  of  corn,  from 
which  Clara  feeds  her  ducks 

and  chickens.     How  many  quarts  of  corn  are  there  for  each  oiie  of 
the  fowls  ? 

21.  How  many  weeks  will  the  corn  last,  allowing  6  quarts  a 
week  for  the  poultry  ?  16. 

22.  A  man  leased  a  farm  for  $865,  at  the  rate  of  $5  an  acre. 
How  many  acres  were  in  the  farm  ?  173. 

3* 


58  INTEGERS. 

23.  A  forwarder  shiiDped  24,744  bushels  of  grain  in  6  equal  car- 
goes.    How  many  bushels  were  in  each  cargo  ?  ^  12 J^. 

24.  A  builder  paid  $5,145  for  bricks,  at  $7  a  thousand.  How 
many  thousand  did  he  buy  ?  735. 

25.  It  cost  $6,504  to  build  a  plank-road  8  miles  long.  What  was 
the  cost  per  mile  ?  $813. 

26.  How  long  will  it  take  a  ship  to  make  a  Toyage  of  889  miles, 
if  she  sails  7  miles  an  hour  ?  127  hours. 

27.  A  boatman  carried  8,532  barrels  of  flour  from  Oswego  to 
New  York  in  9  down  trips.  How  many  barrels  did  he  take  each 
down  trip  ? 

28.  A  party  of  8  men  spent  $1,072  on  a  journey  to  California, 
and  they  shared  the  expense  equally.  How  much  did  each  man 
pay? 

CASE     II. 

The  Divisor  more  than  One  Figure. 

102.  Ex.  Divide  13,091  by  63. 


Explanation. — When  the  divisor  con-  solution. 


sists  of  more  than  one  figure,  the  division  inr 

is  commonly  most   easily  performed  by  

Lonof  Division,  or  the  First  Method  ex-  ^-^^ 
plained  in  Case  I.,  as  shown  in  the  Solu- 


53 

[2J,.7 


tion.  ^71 

103.  It  is   sometimes  difficult  to  tell,  _ - — 

without  trial,  how  many  times  the  divisor  is  contained  in  a 

partial  dividend. 

For  example,  divide  25,474  by  47.  first  trial. 


We  can  not  readily  tell  how  many  times       25  Jf7  Ji. 
47  is  contained  in  254,  but  we  will  sup-       ^^^ 
pose  that  it  is  contained  4  times.     Writ-  6  6 

ing  4  as  the  first  quotient  figure,  we  mul- 
tiply and  subtract,  and  obtain  a  remainder  of  66.     Since 
this  remainder,  which  is  a  part  of  the  partial  dividend,  is 
greater  than  the  divisor,  47  is  contained  in  254  more  than 


DIVISION.  59 


4  times.     We  next  try  5  as  the  quotient        second  tkial. 
figtire  ;  and  the  remainder,  19,  is  less  than      25  Jf7  Jf 


55 


235 
47.     Hence,  47  is  contained  in  254,  5  times. 

We  will  now  suppose  that  47  is  contained         -^  ^  ^ 
in  197,  the  next  partial  dividend,  5  times. 
Writing  5  as  the  second  figure  in  the  quo- 
tient, we  multiply,  and  obtain  a  product  of  235.     Since  this 
product  is  more  than  197,  47  is  not  contained  in  197  as 
many  as  5  times.     Hence, 

I.  When  any  remainder  is  greater  than  the  dwisoVy  the  quo- 
tient figure  is  too  small;  and 

n.  When  any  product  is  greater  than  the  partial  dividend, 
the  quotient  figm^e  is  too  great.      seo  Manual. 

PHOBJOEMS. 

29.  A  drover  paid  $313  for  13  beeves.  How  much  did  they  cost 
him  per  head  ?  $24. 

30.  A  man  on  a  journey  traveled  608  miles  in  19  days.  At  what 
rate  per  day  did  he  travel  ?  33  miles. 

31.  If  21  acres  of  land  produce  945  bushels  of  barley,  what  is  the 
yield  per  acre  ?  4^  iusheU. 

32.  A  farmer  paid  $6,804  for  a  farm  of  108  acres.  What  was  the 
price  per  acre  ? 

33.  A  turnpike-road  46  miles  long  was  kept  in  repair  a  year  at 
■an  expense  of  $1,242.     What  was  the  cost  per  mile  ?  $37. 

34.  In  a  certain  school  the  aggregate  or  total  attendance  for  a 
term  of  65  days  was  7,410.  What  was  the  average  daily  atten- 
dance ?  114. 

35.  How  much  must  a  man  earn  in  each  of  the  313  working  days 
of  a  year,  to  earn  $1,252  in  a  year  ? 

36.  A  dairy-man  packed  10,304  pounds  of  butter  in  56-pound 
tubs.    How  many  tubs  did  he  fill  ?  I84.. 

37.  Last  season  it  cost  a  milkman  $576  to  winter  32  cows. 
What  was  the  cost  per  cow  ?  $18, 

38.  If  1  sheet  of  paper  is  required  for  24  pages  of  a  book,  how 
many  sheets  will  be  required  for  a  book  of  432  pages  ? 


60 


INTEGERS, 


39.  How  many  cars,  each  carrying  49  passengers,  will  be  required 
to  carry  392  passengers  ? 

40.  The  entire  cost  of  constructing  a  railroad  84  miles  long  was 
$4,065,264.    What  was  the  cost  per  mile?  $48,396. 

41.  A  person  whose  property  is  worth  $9,375,  spends  yearly 
$625  more  than  his  income.  In  how  many  years  will  he  spend  all 
his  property  ?  15. 

42.  In  how  many  hours  will  an  express  train  run  the  entire 
length  of  a  railroad  544  miles  long,  running  32  miles  an  hour  ? 

43.  The  stock  of  a  bank,  consisting  of  875  shares,  is  worth 
$117,250.    What  is  the  value  of  each  share  ?  $134. 

4:4:.  A  grain  buyer  shipped  401,400  bushels  of  wheat  from  Mil- 
waukee to  Buffalo  in  24  equal  cargoes.  How  many  bushels  were 
there  in  each  cargo  ?  16,725. 

104.  Ex.  Divide  48,507  by  69. 

Explanation. — In  the  First  Solution  we 
see  that  the  second  remainder,  20,  is  the 
same  as  the  second  partial  dividend. 

In  the  Second  Solution  we  omit  to  mul- 
tiply by  the  quotient  figure,  0,  and  form 
the  third  partial  dividend  by  annexing  7, 
the  next  figure  of  the  dividend,  to  the  20, 
the  second  partial  dividend.     Hence, 

Whenever  the^  partial  dividend  is  less  than 
the  divisor,  ive  place  0  in  the  quotient,  and 
hring  down  the  next  figure  of  the  dividend 
for  a  new  partial  dividend. 

PROBIjEMS, 

45.  How  much  does  a  man  earn  each  month, 
$1,260  a  year  ? 

46.  In  a  factory  275  yards  of  cloth  are  made  daily, 
days  will  be  required  to  make  57,475  yards  ? 

47.  The  total  cost  of  constructing  a  telegraph  line  359  miles 
long  was  $360,795.    What  was  the  cost  per  mile  ?  $1,005. 


FIRST  SOLUTION. 

Jf8507 
Ji.8S 

20 
00 

69 
JOS 

207 
207 

SECOND  SOLUTION. 

U8507 
U8S 

69 
70S 

207 
207 

whose  salary  is 
$105, 

ily.     How 

many 

DIVISION.  61 

48.  The  yield  of  a  Pennsylvania  oil  well  for  1  month  was  30,186 
gallons,  which  was  put  into  casks,  each  holding  43  gallons.  How 
many  casks  were  used  ? 

49.  What  is  the  quotient  of  120,772,144  -v-  592  ?  20J^,007. 

50.  The  dividend  is  524,177,472,  and  the  divisor  5,824.  What  is 
the  quotient?  90,008. 

51.  Divide  1  billion  963  million  198  thousand  504  by  28  thou- 
sand 9.  '^^  thousand  56. 

C^SE     III. 
Remainders  after  Dividing  last  Partial  Dividend. 
105.  Ex.  Affcer  dividing  1,315  acres  of  land  into  as  many 
farms  as  possible  of  92  acres  each,  how  much  land  will  be 
left? 

Explanation. — ^The  divisor,  92,  is  con-  solution. 

tained  in  the  last  partial  dividend,  395,     IS  15  \  92^ 


9  ^ 
4  times,  with  a  remainder  of  27.    As  there     ^  [  1  If. 

is  no  fisfure  of  the  dividend  that  has  not       <5  P  5 

been  used,  we  have  no  figure  to  write       _i_i. 

with  the  27  to  form  a  new  partial  divi-  ^  7'  Remainder. 

dend,  and  the  work  is  completed,  leaving 

a  remainder  of  27.     Hence,  after  dividing  1,315  acres  of 

land  into  14  farms  of  92  acres  each,  there  are  27  acres  re* 

maining  undivided.     See  ManuaL 

FJtOBLJEMS. 

52.  How  many  payments  of  $350  each  must  I  make,  to  pay  for  a 
house  and  lot  worth  $3,400  ?  9,  and  one  payment  of  $250. 

53.  A  farmer  made  963  gallons  of  cider,  which  he  put  into  casks 
holding  41  gallons  each.    How  many  full  casks  had  he  ? 

•    23,  and  19  gallons  over. 

54.  The  dividend  is  80,963,  and  the  divisor  376.  What  is  the 
remainder  ?  123. 

55.  A  forwarder  ships  15,500  barrels  of  flour  to  New  York,  by 
canal.  If  856  barrels  make  a  boat  load,  how  many  full  boat  loads 
has  he  to  ship  ?  18,  and  92  Ixirrels  remainder. 


62 


INTEGERS. 


56.  If  the  directors  of  a  railroad  company  appropriate  $32,000 
for  the  purchase  of  passenger  cars,  and  the  cars  cost  $1,875  each, 
how  many  cars  can  be  bought  with  the  appropriation  ? 

17,  mid  leave  a  surplus  of  $125. 

67.  What  will  be  the  remainder,  after  dividing  62,676  by  573  ? 

58.  How  many  bales  of  396  pounds  each  can  be  made  from 
84,000  pounds  of  cotton  ?  Memainder^  JfS  pounds. 

59.  1,405,169  -=-  3,376  =  how  many  ?  Bemaijider^  953. 

60.  The  dividend  is  5  million  92  thousand  209,  and  the  divisor 
10  thousand  23.     "What  is  the  remainder  ? 

61.  Divide  56,432,782  by  27,541.  Bemainder,  1,273. 


The  Divisor  any  number  of  Tens,  Hundreds,  Thousands, 
and  so  on. 

106.  Ex.  1.  Divide,  67,200  by  100. 

Explanation. — By  the  First  Solution  it  ^^Rst  solution. 
will  be  seen  that,  after  removing  as  many 
figures  from  the  right  hand  of  the  divi- 
dend as  there  are  ciphers  in  the  divisor, 
the  remaining  figures  of  the  dividend  are 
the  same  as  the  quotient.     Therefore, 

In  the  Second  Solution  we  have  brought 
down,  for  the  quotient,  all  the  figures  of  the 
dividend  except  as  many  of  its  right-hand 
figures  as  there  are  ciphers  in  the  divisor. 

Ex.  2.  Divide  49,392  by  1,000. 
Explanation. — By  the  First  Solution  we 
see  that,  if  we  omit  the  three  right-hand 
figures  of  the  dividend,  the  other  figures 
are  the  same  as  the  quotient,  and  that 
the  three  right-hand  figures  thus  omitted 
are  the  same  as  the  remainder.  There- 
fore, 


67200 

100 

600 

672 

720 

700 

200 

200 

8KC0ND  SOLUTION. 

67200 

JOO 

672 


FIK8T  B0LT7TI0IT. 


4.9392 
40  00 

9392 
9000 

392 


1000 
[49 


DIVISION.  63 

In  the  Second  Solution  we  have  brought        second  solxttion. 
down,  for  the  quotient,  all  the  figures  of     Jf9892  \10  0  0 
the  dividend  except  three,  (^.  e.,  as  many     ^P  392 
of  the  right-hand  figures  as  there   are 
ciphers  in  the  divisor),  and  for  the  remainder  we  have 
written  the  three  right-hand  figui'es.     Hence, 

I.  Removing  the  units'  figure  from  a  number,  divides  the 
number  by  10. 

II.  Removing  the  units  and  tens,  or  the  two  right-hand 
figures  from  a  number,  divides  it  by  100. 

III.  Removing  the  three  right-hand  figures  from  a  number, 
divides  it  by  1,000. 

IV.  Removing  the  four  right-hand  figures,  divides  by  10,000  ; 
removing  five  figures,  by  100,000;  removing  six  figures,  by 
1,000,000;  and  so  on. 

V.  Ttie  figures  removed  are  the  remainder,  and  the  other 
figures  are  the  quotient. 

107.  "When  the  divisor  contains  more  than  one  ten,  hun- 
dred, thousand,  etc.,  the  manner  of  obtaining  the  final 
remainder  is  more  difficult.  For  illustrating  the  method, 
we  will  take  the  following  examples : 

(1)  (3)  (3)  (4) 


75  1 15      75_y3_ 
5  25 


15  13      75  [5  I  15  [5      7500  [  100 
5  l5~    [  3  75 


1500  y  100 
15 


5  5  5 

By  carefully  examining  these  four  examples,  we  see  in 
Ex.  1  that  the  quotient  of  75  divided  by  15  is  5  ;  in  Ex.  2, 
3,  that  the  quotient  is  not  changed  by  dividing  both  divisor 
and  dividend  by  3  or  5  ;  and  in  Ex.  4,  that  the  divisor, 
1,500,  and  the  dividend,  7,500,  may  both  be  divided  by  100, 
and  the  results  used  as  divisor  and  dividend,  without  af- 
fecting the  final  quotient.     That  is. 

Both  divisor  and  dividend  may  be  divided  by  the  same  num- 
ber, without  affecting  the  value  of  the  final  quotietit. 


FIU6T  SOLUTION. 


SECOND   SOLUTIOIf. 

SJf.\00 


64  INTEGERS. 

108.  Ex.  Divide  15,284  by  3,400.  -,  r  o  ^  i 

Explanation. — In  the  First  Solution  we  1S600 
divide  as  tauglit  in  Case  III.,  and  obtain  a  iqrJl 
quotient  of  4,  and  a  remainder  of  1,684. 
In  tlie  Second  Solution  we  first  divide- 
botli  divisor  and  dividend  by  100,  (which  i52\8Jf 
we  do  by  cutting  off,  by  a  vertical  line,  13  6 
the  two  right-hand  figures  from  each  IGSA 
term),  obtaining  34  for  a  new  divisor,  152 
for  a  new  dividend,  and  a  remainder  of  84.  Dividing  152 
by  34,  we  obtain  a  final  quotient  of  4,  and  a  second  re- 
mainder of  16.  Since  this  remainder  was  obtained  by  sub- 
tracting 136  hundreds  from  152  hundreds  (see  Eirst  Solu- 
tion), it  must  be  16  hundreds  ;  while  the  first  remainder, 
84,  is  the  tens  and  ones  of  the  dividend.  We  therefore 
unite  these  two  remainders,  by  annexing  the  84  to  the  16 
(hundreds),  and  we  have  1,684  for  the  final  remainder,  the 
same  as  in  the  Eirst  Solution.     Hence, 

In  dividing  by  any  number  of  tens,  hundreds,  thousands, 
and  so  on, 

TJie  final  remainder  consists  of  the  figures  which  were  cut  off 
from  the  dividend,  annexed  to  the  figures  of  the  last  remainder. 

Pit  OBLEMS. 

62.  A  dealer  sold  10  sewing-machines  for  $G50.  How  much  were 
they  apiece  ? 

63.  A  cental  of  grain  is  100  pounds.  How  many  centals  in 
47,300  pounds  ?  Jf73. 

64.  The  capital  or  stock  of  a  certain  mining  company  is 
$235,000,  and  it  is  divided  into  $1,000  shares.  How  many  shares 
of  stock  are  there  ? 

65.  At  $3,000  each,  how  many  steam-tugs  can  be  bought  for 
$6,000  ? 

66.  A  builder  paid  $5,760  for  boards,  at  $30  per  thousand.  How 
many  thousand  feet  did  he  buy  ?  192, 


DIVISION.  65 

67.  A  carriage  maker  received  $1,200  for  light  carriages,  at  $200 
apiece.    How  many  carriages  did  he  sell  ? 

68.  What  is  the  quotient  of  393,400,000  divided  by  100,000  ? 

69.  A  government  agent  paid  $56,400  for  horses  for  the  army,  at 
$150  apiece.    How  many  did  he  buy  ?  376. 

70.  A  farmer  having  $83,  wishes  to  jourchase  yearling  calves,  at 
$10  a  head.     How  many  can  he  buy  ?  8,  and  Jmm  $3  left. 

71.  How  many  freight  cars  will  be  required  to  transport  58,293 
barrels  of  flour,  if  100  barrels  make  one  car  load  ? 

582  full  cars,  and  1  car  carrying  93  larrels. 

72.  A  company  purchase  a  hotel  in  New  Orleans  for  $165,675, 
and  the  payments  are  to  be  $25,000  annually.  How  many  pay- 
ments must  they  make  ?  G  of  $25, 000  each,  and  1  of  $15, 675. 

73.  Divide  103,285  by  36,000.  Memainder,  13,285. 

74.  What  is  the  quotient  of  17,630,000  divided  by  24,000  ? 

109.  The  facts  deduced  in  Arts.  95,  98,  106, 107,  may  now 
be  stated  as 

General  ^rmciples  of  2)lrisio?i, 

I.  A  concrete  number  can  he  divided  by  either  a  conci^ete  or 
an  abstract  number. 

II.  An  abstract  number  can  be  divided  by  an  abstract  number 
only. 

III.  The  quotient  is  an  abstract  number ^  when  the  divisor  and 
dividend  are  both  abstract  or  both  concrete  numbers. 

rV.  The  quotient  is  a  concrete  number,  when  the  divisor  is  an  , 
abstract,  and  the  dividend  a  concrete  number. 

V.  Any  quotient  figure  is  of  the  same  order  of  units  as  the 
last  figure  of  the  dividend  used  to  obtain  it. 

VI.  2he  removal  of  the  right-hand  figure  from  a  number 
divides  that  number  by  10. 

VH.  Both  divisor  and  dividend  may  be  divided  by  the  same 
number,  without  affecting  the  value  of  the  final  quotient. 

Vm.  The  right-hand  figure  of  any  remainder  is  of  the  same 
order  of  units  as  the  last  figure  of  the  dividend  used. 


66  INTEGERS. 

llOt  Upon  these  principles  is  based  the 

2iule  for  2)lvlsio7i  of  Integers, 
I.  For  Long  Division. 
1.  Flace  the  divisor  at  the  right  of  the  dividend,  separate 
them  by  a  line,  and  draw  a  line  under  the  divisor  to  separate  it 
from  the  quotient. 

2.-  Find  how  many  times  the  divisor  is  contained  in  the  first 
partial  dividend,  and  write  the  result  for  the  first  figure  of  the 
quotient. 

3.  Multiply  the  divisor  by  this  quotient  figure,  suUract  the 
product  from  the  partial  dividend  used,  and  to  the  remainder 
annex  the  next  figure  of  the  dividend  for  a  new  partial  divi- 
dend. 

4.  In  the  same  manner,  continue  to  divide,  multiply,  suh- 
tract,  and  hring  down,  until  all  the  figures  of  the  dividend 
have  teen  used, 

II.  For  Short  Division. 

1.  Write  the  divisor  and  dividend  as  in  Long  Division,  and 
draw  a  line  under  the  dividend  to  separate  it  from  the  quotient, 

2.  Mnd  hoio  many  times  the  divisor  is  contained  in  the 
first  partial  dividend,  as  in  Long  Division,  and  write  the 
result  under  the  last  figure  of  the  dividend  used,  for  the  first 
figure  of  the  quotient, 

3.  Multiply,  subtract,  and  form  a  neiu  partial  dividend,  as 
in  Long  Division,  performing  the  operations  mentally. 

4.  Divide  this  partial  dividend,  and  write  the  result  as  the 
second  figure  of  the  quotient. 

5.  Proceed  in  the  same  manner  until  all  the  figures  of  the 
dividend  have  been  used, 

in.  For  dividing  by  any  number  of  tens,  hundreds,  thou- 
sands, and  so  on. 
1.  Gut  off  the  ciphers  by  a  line,  and  also  an  equal  number 
of  figures  from  the  right  oftlie  dividend. 


DIVISION.  67 

2.  Divide  the  remaining  figures  of  tlie  dividend  ly  the  re- 
maining figures  of  the  divisor. 

3.  For  the  true  remainder,  annex  to  the  last  remainder  the 
figures  cut  off  from  the  dividend. 

JPJt  OBIjEMS. 

75.  An  army  contractor  paid  $276,560  for  beef,  at  $16  a  barrel. 
How  much  beef  did  he  buy  ?  17,285  barrels. 

76.  In  a  cotton-factory  are  54  looms,  which  were  bought  at  a 
total  cost  of  $9,720.     What  was  the^cost  of  each  loom  ? 

77.  A  canal  97  miles  long  was  constructed  at  a  cost  of  $14,131,930. 
What  was  the  cost  per  mile  ?  $14^j  690. 

78.  Peter  having  an  ear  of  corn  in  which  were  864  kernels, 
planted  it  in  hills  of  6  kernels  each.   How  many  hills  did  he  plant  ? 

79.  He  planted  the  corn  in  8  equal  rows.  How  many  hills  were 
there  in  each  row  ?  18. 

80.  The  fare  of  219  passengers  by  steam-ship  from  New  York  to 
Havre,  was  $36,135.     How  much  was  the  fare  of  each  passenger  ? 

81.  At  $60  a  head,  how  many  cows  can  be  bought  for  $1,650  ? 

27,  with  $30  left. 

82.  A  city  builder  received  $780,000  for  building  brown-stone 
front  houses,  at  an  average  price  of  $10,000  each.  How  many 
houses  did  he  build  ? 

83.  A  manufacturer  sold  reapers  at  $130  each,  and  received 
$6,240.     How  many  reapers  did  he  sell  ?  4^. 

84.  How  many  barrels,  each  holding  200  pounds,  will  be  required 
for  packing  47,875  pounds  of  pork? 

239,  with  75  pounds  of  porTo  left. 

85.  A  wholesale  grocer  bought  3,440  pounds  of  tea,  in  80-pound 
chests.     How  many  chests  did  he  buy  ?  J^. 

86.  How  many  canal  boats  can  a  transportation  company  buy 
with  $34,000,  at  $1,000  each  ? 

87.  In  the  schools  of  a  certain  city  28,497  pupils  are  taught  by 
483  teachers.     What  is  the  average  number  of  pupils  to  a  teacher  ? 


68  INTEGERS. 

88.  A  paper  maker  having  361,920  sheets  of  foolscap,  put  it  up 
for  market  in  quires  of  34  sheets  each.  How  many  quires  were 
there  ? 

89.  He  sold  the  paper  by  the  ream  of  30  quires.  How  many 
reams  did  he  sell  ?  75^., 

90.  The  United  States  Government  paid  $103,600  for  740  army 
wagons.     How  much  was  that  for  each  wagon  ?  $1^0. 

91.  What  is  the  quotient  of  447  billion  670  million  621  thou, 
sand  104  divided  by  4  million  930  thousand  76  ?     90  tJiousaml  8O4. 

92.  Divide  660,886,723  by  982.  Remainder,  723. 

93.  The  dividend  is  468,002,659,  and  the  divisor  9,497.  What 
is  the  remainder  ?  9^,493. 

94.  A  planter  raised  86,301  pounds  of  cotton  on  233  acres  of 
land.     How  many  pounds  was  the  yield  i)er  acre  ?  387. 

95.  The  New  York  and  New  Haven  Railroad  track  is  401,380 
feet  long.  How  many  miles  from  the  New  York  to  the  New 
Haven  Railroad  depot,  there  being  5,380  feet  in  a  mile  ?  76. 

96.  In  a  certain  county  $3,039,688  were  paid  in  bounties  to 
3,394  volunteers.    What  bounty  was  paid  to  each  soldier  ?     $852. 

97.  How  long  will  564,000  rations  last  an  army  brigade  of  5,875 
men  ?  96  days. 

98.  A  produce  buyer  purchased  417  bushels  of  wheat,  873  bush- 
els of  oats,  and  314  bushels  of  barley.  The  bins  in  his  storehouse 
will  hold  73  bushels  each.  How  many  bins  will  each  kind  of 
grain  fill  ?  Wheat,  5  Una,  and  57  bushels  over. 

Oats,     12    "■      "      9      "         " 
Barley,  ^    "       "    26      "        " 

99.  If  all  of  the  grain  was  of  one  kind,  how  many  bins  would  it 
fill  ?  ,  20  hashels,  remainder. 

100.  A  man  buys  a  farm  of  113  acres,  at  $54  an  acre.  He  pays 
$1,392  down,  and  agrees  to  pay  the  balance  in  6  equal  yearly  pay- 
ments.   How  much  of  the  debt  must  he  pay  each  year  ?       $785. 


REVIEW    PROBLEMS.  69 

SECTION  VII. 
^:^riByr  i>^osi.bms  ij\r  ijytbgb^rs, 

1.  The  parts  of  a  number  are  73,  437,  856,  and  32,519.   What  is 

the  number  ?        See  Manual. 

2.  The  minuend  is  59,408,  and  the  subtrahend  14,642.  What  is 
the  remainder  ? 

3.  The  sum  of  two  numbers  is  1  million  56  thousand,  and  one  of 
the  numbers  is  304  thousand  9.     What  is  the  other  number  ? 

4.  A  reward  of  $7,650  was  shared  among  4  detectives,  the  first 
of  whom  received  $2,225,  the  second  $1,750,  and  the  third  $1,875. 
How  much  did  the  fourth  receive  ? 

5.  The  multiplicand  is  185,046,  and  the  multiplier  4,309.  What 
is  the  product  ? 

6.  What  is  that  number,  the  factors  of  which  are  384,  27, 90,  and 
10,000  ? 

7.  The  dividend  is  1,728,000,  and  the  divisor  1,800.  What  is  the 
quotient  ? 

8.  If  the  dividend  is  5,443,200,  and  the  several  successive  divisors 
are  9,  14,  and  600,  what  is  the  final  quotient  ? 

9.  A  farmer's  expenses  and  receipts  one  year  were  as  follows : 


^oi    7r/Cea-/. 

'^     ^a'/<i ^S'S 


c^i   ^aifoi 0PS 

^'     S'eec/ ^P 

■men^ /J^ 

^^     (Jk'^ei&d'^  ty/^oney. ///O 

Did  he  make  or  lose  money  that  year,  and  how  much  ?         $385. 

10.  He  sold  his  hay  at  $8  a  ton.     How  much  hay  did  he  sell  ? 

11.  If  7  bricklayers  are  67  days  in  putting  up  the  walls  of  a 
machine-shop,  and  each  man  lays  1,950  bricks  a  day,  how  many 
bricks  will  there  be  in  the  walls  of  the  building  ?  9H,550. 


7a-?/ 


//f 

niZ /// 


"Mo/. ps 


70  INTEGERS. 

13.  From  a  cistern  containing  19,437  gallons  of  water,  13,294  gal» 
Ions  were  drawn  out,  and  afterward,  during  a  rain  storm,  7,483  gal- 
lons ran  in.     How  much  water  was  there  in  the  cistern  ? 

13,626  gallons. 

13.  The  live  weight  of  an  ox  was  1,816  pounds.  When  dressed, 
the  four  quarters  weighed  respectively  271,  264,  275,  and  287 
pounds ;  the  hide  weighed  85  pounds,  and  the  tallow  97  j)ounds. 
What  was  the  diflference  between  the  live  and  dead  weight  of 
the  ox?  537 pounds. 

14.  One  season  a  jobbing  carpenter  built  5  dwellings,  which  cost 
him  $3,176,  $5,194,  $1,342,  $6,950,  and  $788.  He  received  for 
building  them  $3,875,  $6,820,  $1,280,  $7,896,  and  $875.  What 
were  his  season's  profits  ?  $3,296. 

15.  The  cost  of  mounting  and  equipping  a  cavalry  regiment  of 
1,037  men  was  $213,662.     How  much  was  the  cost  per  man  ? 

16.  One  year  a  stove  manu-  53  No.  10  Stoves,  at  $33 
facturer  sold  to  a  wholesale 
dealer,  stoves  as  i3er  annexed 
schedule. 
.   To  how  much  did  the  sales  73   "      6      "       "     18 


169    " 

9 

2U   " 

S 

192   " 

7 

73   " 

6 

amount  i       See  Manual.  ^j^y  ^i^Q 

17.  I  bought  a  farm  of  153  acres,  at  $95  an  acre,  and  paid  down 
$2,500.     How  much  of  the  purchase  money  remained  unpaid  ? 

18.  A  man  bequeathed  to  each  of  2  sons  $7,600 ;  to  a  third  son 
$1,500 ;  to  each  of  3  daughters  $3,775 ;  and  the  balance  of  his 
estate,  which  amounted  to  $6,877,  to  other  parties.  But  the  will 
was  set  aside,  and  the  property  was  divided  equally  among  his 
children.     How  much  did  each  receive  ?  $5,817. 

19.  A  merchant  bought  a  piece  of  broadcloth  containing  56 
yards,  for  $266,  and  sold  it  at  $6  a  yard.  How  much  was  his 
profit  ? 

20.  In  the  Oakland  Mill  are  9  run  of  stone,  each  capable  of 
grinding  100  bushels  of  wheat  per  day.  In  what  time  can  297,000 
bushels  of  wheat  be  ground  ?  330  days. 

21.  A  dairy-man  has  fodder  enough  to  keep  his  85  cows  4  months. 
If  he  sells  7  cows,  how  many  months  will  the  fodder  last  the  rest  ? 


'^'/o/ZS'Mu^J,  a'/^SS,  /_ 


S^ 

// 

-  ^^  __. 

S2 

// 

''    22,     ___ 

/^ 

// 

^^    /^     ___ 

cf 

// 

^^    /i',     ___ 

cf 

// 

REVIEW    PROBLEMS.  71 

22.  The  receipts  and  expenditures  of  a  church  society  for  one 
year  were  as  follows  : 

9.  Expenditures. 

''   ^/ol^'/ei JOO 

''         ''   Ma;'/on ^/2 

(SoT/iendeJ   /oi  ci^ue/- //'-t' 

^^     ^y^'/, PJ 

<^iee,           (S'      ^^                                           ^an'/fk^en'/  Gdp/ien^ed /2/ 

How  did  the  account  stand  at  the  close  of  the  year  ? 

23.  The  United  States  Supreme  Court  consists  of  a  Chief  Justice, 
whose  salary  is  $6,500  per  annum,  and  9  associate  justices,  whose 
salaries  are  $6,000  per  annum  each.  How  much  do  all  their  sal- 
aries amount  to  in  one  Presidential  term  ?  $2^2,000, 

24.  How  many  military  companies  of  98  men  each  can  be  formed 
from  3,675  recruits  ?  37,  with  40  recruits  left. 

25.  A  wood  dealer  sold  36  cords  of  hickory  wood,  at  $6  a  cord, 
75  cords  of  maple  wood,  at  $5  a  cord,  and  43  cords  of  soft  wood, 
at  |4  a  cord.     How  much  did  he  receive  for  the  whole  ?       $76S. 

26.  A  farmer  who  raised  984  bushels  of  oats,  after  retaining  48 
bushels  for  seed,  and  enough  to  winter  5  horses,  allowing  50  bush- 
els to  each  horse,  sold  the  balance.  How  many  bushels  did  he 
sell  ?  686. 

27.  A  lady  having  $100,  paid  $58  for  a  set  of  furs,  and  $2  a 
yard  for  17  yards  of  silk.     How  much  money  had  she  left  ?      $8. 

28.  A  grocer  bought  2,880  pounds  of  coffee  in  120-pound  sacks. 
How  many  sacks  did  he  buy  ? 

29.  A  grocer  bought  5  hogsheads  of  molasses  that  were  billed  to 
him  at  140  gallons  each ;  but  the  first  was  17  gallons  short,  the 
second  5  gallons,  the  third  9  gallons,  the  fourth  4  gallons,  and  the 
fifth  2  gallons.     How  many  gallons  were  in  the  5  hogsheads  ? 

30.  A  soldier  enlisting  for  3  years  received  a  bounty  of  $949. 
He  served  8  months  as  a  j^rivate,  at  $13  a  month;  8  months  as  a 
corporal,  at  $14  a  month;  13  months  as  a  sergeant,  at  $17  a 
month ;  and  7  months  as  an  orderly  sergeant,  at  $18  a  month. 
What  was  the  total  amount  of  his  pay  ?  How  much  did  it  average 
per  month  ?      See  Manual.  $42  per  month. 


72 


INTEGERS. 


Table  of  Areas  used  in  the  next  ten  Problems. 


SQUARE  MILES. 

United  States 3,001,003 

France 207,829 

England 50,922 

Island  of  Australia. .  .2,980,770 
"       "   Borneo  (about)    320,000 

"       "   Cuba 47,278 

Texas 280,000 

California 159,000 


SQUARE  MILES. 

New  England 65,038 

Michigan 56,243 

Illinois 55,405 

Georgia 52,009 

New  York 50,519 

Tennessee 45,600 

Ohio 39,964 

Massachusetts 7,800 

See  ManuaL 

31.  Into  how  many  states,  each  as  large  as  New  York,  could 
Australia  be  divided  ?  59 ,  and  1^9  square  miles  remainder. 

32.  How  many  states,  each  as  large  as  Massachusetts,  might  be 
formed  from  Texas  ? 

33.  The  Island  of  Borneo  is  how  many  times  as  large  as  Cuba  ? 

34.  How  much  larger  is  the  State  of  Illinois  than  England  ? 

35.  The  Island  of  Australia  is  how  many  times  as  large  as  New 
England  ?  and  how  much  larger  or  smaller  than  Tennessee  is  the 
remainder  ?  8,^60  square  miles. 

36.  If  all  the  territory  of  the  United  States  were  divided  into  as 
many  states  as  possible,  each  as  large  as  Michigan,  the  remainder 
forming  another  state,  how  many  states  would  there  be  in  the 
Union,  and  what  would  be  the  size  of  the  small  state  ? 

5j^  states  ;  20, 123  square  miles. 

37.  If  a  country  consisted  of  376  states,  each  as  large  as  Massa- 
chusetts, would  its  area  be  greater  or  smaller  than  the  area  of  the 
United  States  ?  68,202  square  miles. 

38.  If  Texas  were  divided  into  4  states  of  equal  size,  how  much 
larger  than  Ohio  would  each  of  them  be  ? 

39.  How  does  3  times  the  area  of  Georgia  compare  with  the  area 
of  California  ? 

40.  Into  how  mcny  countries  could  the  United  States  be  divided, 
and  each  have  an  area  equal  to  the  combined  areas  of  France  and 
England  ?  154,741  square  miles  more  tJian  11  such  countries. 


SECTION  I. 

111.  We  have  already  learned  (Chap.  I.,  Sec.  II.)  that  the 
places  of  the  different  orders  of  integral  units— as  ones, 
tens,  hundreds,  thousands,  ten-thousands,  and  so  on — 
increase  in  valae  from  right  to  left  in  a  tenfold  ratio,  or  by 
the  constant  multipher  10.  Thus,  10  times  ones  are  tens, 
10  times  tens  are  hundreds,  10  times  hundreds  are  thou- 
sands, and  so  on. 

We  have  also  learned  that  the  orders  of  units  decrease 
from  left  to  right  in  a  tenfold  ratio,  or  by  the  constant 
divisor  10.  Thus,  thousands  -^  10  are  hundreds,  hundreds 
-^  10  are  tens,  tens  -^  10  are  ones,  or  simple  units. 

112.  Continuing  this  division  below  ones,  we  obtain  a 
new  class  of  numbers,  which  are  subject  to  the  same  gen- 
eral laws  as  are  integers,  and  which  differ  from  them  in 
only  one  respect,  namely  ;  the  value  of  a  unit  of  any  order 
is  less  than  unity,  or  1.     Thus, 


If  ones           are  divided  by  10,  the  resulting 

units 

are  tenths  ; 

"  tenths                    " 

10, 

(i 

"    hundredths; 

"  hundredths            " 

10, 

(( 

"    thmisandths ; 

"  thousandths          " 

10, 

u 

"    ten-t/iomandths  ; 

"ten-thousandths    " 

10, 

u 

"   hundred-thousandths 

"  hundred-thousandths 

10, 

(( 

"   miUionths; 

and  so  on. 

113#  ^  Scale y  in  Arithmetic,  is  an  established  order  of 
increase  or  decrease  from  any  unit  to  higher  or  lower  units 
in  the  same  class  of  numbers. 
4 


74  DECIMALS. 

114.  A  ^eciniat  Scale  is  one  in  which  the  values  of 

the  orders  of  units  increase  by  the  constant  multiplier,  and 

decrease  by  the  constant  divisor,  10. 

Notes.— 1.  The  term  decimal  is  derived  from  the  Latin  decern^  wMch  sig- 
nifies 10. 
2.  The  scale  of  integers  is  a  decimal  scale. 

115.  A  Decimal  LP?iit  is  one  of  the  equal  decimal 
parts  into  which  a  thing,  or  the  unit  1,  is  divided  ;  as, 
1  tenth,  1  hundredth,  1  thousandth,  1  ten-thousandth,  and 
so  on. 

116.  A  Decimal  is  a  number  expressed  by  decimal 
units  ;  as,  7  tenths,  258  thousandths. 

Notes.— 1.  A  number  consisting  of  an  integer  and  a  decimal  is  a  Mixed 
Number ;  as,  8  and  25  hundreds. 

2.  Integers  and  decimals  together  form  one  general  class  of  numbers, 
called  Decimal  Numbers. 


SECTION  II. 

JVOTATIOJSr  AJV3)  JVUMB^ATIOJV, 

117.  If  we  divide  an  apple  into  10  equal  parts,  each  of 
the  parts  is  1  tenth  of  the  apple.  When  any  thing,  or  a  1, 
is  divided  into  10  equal  parts,  1  of  the  parts  is  1  tenth  of  the 
thing  or  the  1,  2  of  the  parts  are  2  tenths^  3  of  them  are 
3  tenths,  4  of  them  are  Jf  tenths,  and  so  on.     10  tenths  are  1. 

1  tenth  is  written  .1 


2  tenths  are  written  .2 

3  "         "         "       .3 

4  "        "        "       .Jf 

5  "        "        "      .5 


6  tenths  are  written  .6 

ly        u  a  ((  fy 

8  "  "  "         .8 

9  "         "         "       .9 


118.  In  the  number  111  the  first  or  left-hand  figure  is  1 
hundred,  the  second  figure  is  1  ten,  and  the  third  figure  is 
1  one.  Since  1  ten  is  1  tenth  of  1  hundred,  and  1  one  is 
1  tenth  of  1  ten,  it  follows  that 


NOTATION     AND     NUMERATION.  75 

The  value  of  any  figure  in  a  number  is  1  tenth  of  the  value 
of  a  like  figure  standing  in  the  next  place  at  the  left.     Hence, 

119.  The  value  of  any  figure  written  at  the  right  of  ones 
is  tenths. 

1  and  1  tenth  are  written    1.1 

19  and  3  tenths   "         "        19.3 

50  and  7  tenths   "         "        50.7 

276  and  9  tenths  "         "      276.9 

120.  The  decimal  ^oint  is  a  period  or  point  (.) 
placed  before  tenths.  When  placed  between  figures,  the 
decimal  point  is  always  read  and.     Thus,  4.5  is  read  "  4  and 

5  tenths."      see  Manual. 

In  writing  decimals,  the  decimal  point  must  always  be 
used. 

EXEJRCISES. 

1.  Read  .4,  .8,  .1,  7.3,  10.9,  393.6,  7198.3. 
3.  Write  five  tenths,  one  tenth,  nine  tenths. 

3.  Write  17  and  3  tenths;  38  and  6  tenths. 

4.  Write  340  and  9  tenths;  1006  and  5  tenths. 

5.  Write  two  tenths  ;  five  hundred  sixty  and  four  tenths. 

121.  If  any  thing,  or  a  1,  be  divided  into  tenths,  and 
then  each  of  the  tenths  be  divided  into  10  equal  parts,  there 
will  be  in  the  whole  thing,  or  the  1,  10  times  10,  or  100 
equal  parts  ;  and  each  of  the  parts  will  be  1  hundredth  of 
the  whole  thing,  or  of  the  1.     Hence, 

1  hundredth  is  1  tenth  of  1  tenth.  And,  since  the  value  of 
a  figure  in  any  place  is  1  tenth  of  the  value  of  a  like  figure 
in  the  next  place  at  the  left  (see  118),  it  follows  that 

The  value  of  any  figure  written  at  the  right  of  tenths  is 
hundredths. 

1  hundredth  is  written  .01 
3  hundredths  are  "  .02 
5  "  "      "        .05 

8  "  "      "       .08 


76  DECIMALS. 

122.  .37  is  3  tenths  and  7  hundredths.  But  3  tenths  = 
30  hundredths,  and  30  hundredths  +  7  hundredths  =  37 
hundredths. 

Tenths  and  hundredths  are  read  together  as  hundredths. 

,23,  or  3  tenths  and  3  hundredths,  is  read  23  hundredths. 
.57,  or  5  tenths  and  7  hundredths,  is  read  57  hundredths. 
6.85,  or  6  ones,  8  tenths,  and  5  hundredths,  is  read  6  and  85  hun- 
dredths. 

6.  Read  .43,  .91,  .04,  8.32,  5.09,  47.47,  5080.0G. 

7.  Write  3  hundredths;  51  hundredths;  2  and  75  hundredths. 

8.  Write  15  and  15  hundredths ;  328  and  11  hundredths. 

9.  Write  30  and  30  hundredths ;  200  and  2  hundredths. 

123.  If  any  thing,  or  a  1,  be  divided  into  hundredths, 
and  then  each  of  the  hundredths  be  divided  into  10  equal 
l^arts,  there  will  be  in  the  whole  thing,  or  the  1,  100  times 
10,  or  1,000  equal  parts  ;  and  each  of  the  parts  will  be  1 
thousandth  of  the  whole  thing,  or  of  1.     Hence, 

1  thousandth  is  1  tenth  of  1  hundredth.  And  the  value  of 
any  figure  written  at  the  right  of  hundredths  is  thou- 
sandths.    (See  118). 

1  thousandth    is  written  .001 

4  thousandths  are      "        .00^ 

7  "  "        ''        .007 

9  "  "        "         .009 

124.  .278  is  2  tenths  7  hundredths  and  8  thousandths,  or 
27  hundredths  and  8  thousandths.  But  27  hundredths  = 
270  thousandths,  and  270  thousandths  +  8  thousandths  = 
278  thousandths. 

Tenths,  hundredths,  and  thousandths  are  read  together  as  thou- 
sandths. 

.006  is  read  6  thousandths. 
.072  is  read  72  thousandths. 
.493  is  read  493  thousandths. 
19.136  is  read  19  and  136  thousandths. 


NOTATION    AND    NUMERATION.  77 

EXJEJtCISJES. 

10.  Eead  .43,  .18,  .03,  6.27,  343.51,  99.07. 

11.  Eead  .170,  .584,  .096,  .304,  .007,  .901. 

13.  Eead  4.33,  19.07,  70.219,  9.031,  317.108,  11.005. 

13.  Write  5  tenths  and  6  hundredths,  or  56  hundredths. 

14.  Write  93  hundredths  ;  6  hundredths. 

15.  Write  1  tenth  9  hundredths  and  7  thousandths,  or  197  thou- 
sandths. 

16.  Write  311  thousandths  ;  42  thousandths. 

17.  Write  three  hundred  seven  thousandths. 

18.  Write  30  and  19  hundredths. 

19.  Write  356  and  4  hundredths. 

30.  Write  193  and  5  thousandths. 

31.  Write  3,281  and  59  thousandths. 
23.  Write  10,000  and  308  thousandths. 

125.  A  figure  at  the  riglit  of  thousandths  is  ten-thou- 
sandths; and  a  decimal  containing  tenths,  hundredths,  thou- 
sandths, and  ten-thousandths,  is  read  as  ten-thousandths. 

.5763  is  5  tenths,  7  hundredths,  6  thousandths,  and  3  ten-thou- 
sandths, and  is  read  5,763  ten-thousandths. 

126.  A  figure  at  the  right  of  ten-thousandths  is  hundred- 
thousandths  ;  and  a  decimal  containing  tenths,  hundredths, 
thousandths,  ten-thousandths,  and  hundred-thousandths,  is 
read  as  hundred-thousandths. 

.57308  is  5  tenths,  7  hundredths,  3  thousandths,  0  ten-thou- 
sandths, and  8  hundred-thousandths,  and  is  read  57,308  hundred- 
thousandths. 

127.  A  figure  at  the  right  of  hundred-thousandths  is 
millionths,  a  figure  at  the  right  of  millionths  is  ten-millionths, 
a  figure  at  the  right  of  ten-milHonths  is  hundred-millionlhSj 
and  so  on. 

128.  When  the  right-hand  figure  of  a  decimal  is  mill- 
ionths, the  whole  decimal  is  read  as  millionths  ;  when  the 
right-hand  figure  is  ten-millionths,  the  decimal  is  read  as 
ten-millionths  ;   when  the  right-hand  figure  is  hundred- 


78  DECIMALS. 

millionths,  the  decimal  is  read  as  hundred-millionths,  and 
so  on.     And  in  general, 

The  figures  of  a  decimal  are  read  the  same  as  the  figures  of 
an  integer,  and  to  the  whole  decimal  is  given  the  local  name 
of  the  last  or  right-hand  figure. 

.47^298  is  476,298  milliontlis. 

.5008721  is  5,008,721  ten-millionths. 

.87396483  is  87,396,483  hundred-milliontlis.  ,      - 

.000084  is  84  millionths. 

.0005008  is  5,008  ten-milliontlis. 

.08070802  is  6,070,802  hundred-millionths. 

129.  .5  is  5  tenths.  .50  is  50  hundredths,  or  5  tenths 
and  0  hundredths.  .500  is  500  thousandths,  or  5  tenths, 
0  hundredths,  and  0  thousandths.  That  is,  .5,  .50,  and  .500 
are  all  of  the  same  value,  namely,  5  tenths  ;  consequently, 
ciphers  on  the  right  of  a  decimal  do  not  change  the  places 
of  the  other  figures.     Hence, 

I.  Ciphers  mag  le  annexed  to  any  decimal,  or  decimal 
ciphers  to  any  integer,  without  changing  its  value;  and 

n.  Ciphers  may  he  omitted  from  the  right  of  any  decimal, 
or  decimal  ciphers  from  the  right  of  any  integer,  without 
changing  its  value. 

130t    TABLE   OF   VALUES   OF   DECIMAL  NUMBERS. 

One  decimal  figure  expresses  tenths. 
Two  decimal  figures  express  hundredths. 


Three    " 

thousandths. 

Four      " 

ten-thousandths. 

Five       " 

hundred-thousandths. 

Six 

millionths. 

Seven     " 

ten-millionths. 

Eight     " 

hundred-millionths. 

Nine      " 

u 

Mllionths. 

And  so  on. 

See  Manual. 

NOTATION     ANDNUMERATION.  79 

131  •  Figures  standing  in  places  at  equal  distances  to  the 
right  and  left  of  ones  have  names  that  correspond  to  each 
other,  as  shown  in  the  following 

DIAGRAM   OF  DECIMAL   NOTATION. 

9876543  2  1.  2345G789 


132.  This  diagram  also  shows  that  the  value  which  any 
figure  in  a  decimal  expresses,  is  determined  by  the  place  it 
occupies. 

133.  From  the  illustrations  and  explanations  now  given, 
we  deduce  the  following 

Principles  of  decimal  JVotation  and  JVumeration, 

I.  All  places  to  the  right  of  unity  are  decimals. 

n.  The  values  of  the  different  places  in  a  decimal  increase 
from  right  to  left,  and  decrease  from  left  to  right,  in  a  tenfold 
ratio. 

m.  The  place  which  any  figure  occupies  in  a  decimal  de- 
termines the  value  expressed  hy  it  in  that  decimal. 

rV.  The  decimal  point  must  alivays  he  placed  before  tenths. 

V.  In  writing  a  decimal,  all  places  not  named  must  he 
filled  ly  ciphers. 

VI.  Decimal  ciphers  may  he  annexed  to,  or  omitted  from, 
the  right  of  any  number,  luithout  changing  its  value. 

VII.  The  names  of  places  equi-distant  to  the  left  and  right 
of  unity  differ  only  in  their  terminations,  those  at  the  left 
terminating  in  ns  or  ds,  and  those  at  the  right  in  ths. 

Vni.  The  figures  of  a  decimal  are  read  the  same  as  the 


80  DECIMALS. 

figures  of  an  integer,  the  name  of  the  place  occupied  ly  the 
right-hand  figure  of  the  decimal  leing  pronounced  after  the 

last  figure  read.        see  Manual. 

131*  These  principles  fully  estabHsh  this 

General  Zaw  of  3)ecimaZ  JVumbers, 

Integers  and  decimals  form  one  class  of  numbers,  in  the 
decimal  scale  ;  and  all  like  operations  upon  the  tivo  divisions 
of  the  class  are  governed  ly  the  same  pri7iciples. 

EXEM  CIS  ES. 

23.  Read  .1765 ;  .3046 ;  92.1005  ;  100.0048. 

24.  Eead  .39417 ;  .00009 ;  53.40206 ;  10.00538. 

25.  Read  .476398  ;  11.000141 ;  904.204080  ;  21.600008. 

26.  Read  .4598217;  19.3006009;  214.0380965. 

27.  Read  .00000001 ;  .70876941  ;   329000.80000185. 

28.  Read  3976.070009  ;  56.0085  ;  10006.000596. 

29.  Read  5.5682;  273.8760099;  1.000000007. 

80.  Write  291  ten-thousandths ;  write  706,095  millionths. 

31.  Write  508  millionths ;  write  217  hundred-millionths. 

32.  Write  90,085,765  hundred-millionths. 

33.  Write  5  ten -millionths  ;  write  18  ten-billionths. 

34.  Write  3,750  and  17  ten-thousandths. 

35.  Write  7  thousand  and  7  thousandths. 

36.  Write  2,548,006  and  905  millionths. 

37.  Write  19  and  19billiontlis. 

38.  Write  297,641,879  trillionths ;  write  700,849  ten-billionths. 

39.  Write  six  hundred  seventeen  millionths. 

40.  Write  six  hundred  and  seventeen  millionths. 

41.  Write  four  thousand  seven  hundred-thousandths. 

42.  Write  four  thousand  and  seven  hundred-thousandths. 

43.  Write  four  thousand  and  seven  hundred  thousandths. 

44.  Write  eight  hundred  nine  thousand  one  hundred  fifty-seven 
ten-millionths. 

45.  Write  sixty-three  million  three  hundred  fifty-four  thousand 
eight  hundred  seventy-seven  billionths. 

46.  Write  one  thousand  four  hundred  and  ten  thousandths. 

See  Manual 


ADDITION.  81 

SECTION   III. 
^  !Z>  ^  z  r  z  o  jv. 

135.  Ex.  What  is  tlie  sum  of  56.125,  9.356,  and  123.25  ? 

Explanation.  —  Since   only  like    orders  of  solution. 

units  in  different  numbers  can  be  added  (see  5  6.125 

89,  II.),  we  write  the  numbers  with  like  orders  y  ^'i't'i 
of  units — both  decimal  and  inteoral — in  the 


same  columns.  The  decimal  points  then  stand  1-88.731 
in  a  column.  We  commence  at  the  right,  and 
add  as  in  integers.  Since  the  sum  of  thousandths  is  thou- 
sandths, the  sum  of  hundredths  is  hundredths,  and  the  sum 
of  tenths  is  tenths,  and  there  are  thousandths,  hundredths, 
and  tenths  in  the  given  parts,  there  must  also  be  thou- 
sandths, hundredths,  and  tenths  in  their  sum.  We  therefore 
place  the  decimal  point  in  the  sum  before  the  7,  and  directly 
under  the  decimal  points  in  the  parts. 

136.    ^ule  for  oiddltlon  of  decimals, 

I.  Write  the  numbers  so  that  the  decimal  points  shall  stand 
in  a  column. 

II.  Add  in  the  same  manner  as  i7i  integers,  and  place  the 
decimal  point  in  the  sum  directly  under  the  decimal  points  in 

the  parts.       see  Manual. 

PM  OBLEMS. 


(1) 

(2) 

(3) 

(4) 

(5) 

.331 

.48 

3.62 

162.71 

.0052 

.746 

.697 

517.83 

48.086 

.02081 

.984 

.8 

21.9 

3915.3004 

.016 

.258 

.5764 

674.08 

■  .721 

.0000375 

6.  A  farmer  brought  three  loads  of  wood  to  market,  the  first  load 
containing  .8  of  a  cord,  the  second  .75,  and  the  third  .9375.  How 
many  cords  did  he  bring  in  all  ?  2.^875. 

4* 


82  DECIMALS. 

7.  A  peddler  traveled  6.75  miles  one  day,  4.6  miles  the  next, 
7.384  miles  the  third,  and  2.14  miles  the  fourth.  How  far  did  he 
travel  in  the  four  days  ?  20.87 Jf,  miles. 

8.  How  many  acres  in  four  ^elds,  there  being  9.5  acres  in  the 
first,  11.4  acres  in  the  second,  8.75  acres  in  the  third,  and  12.675 
acres  in  the  fourth  ?  Jt2.325. 

9.  A  lady  bought  16.25  yards  of  silk,  12.75  yards  of  alpaca,  6.5 
yards  of  merino,  11.875  yards  of  delaine,  and  23.5  yards  of  French 
calico.     How  many  yards  of  goods  did  she  purchase  ?        70.875. 

10.  A  farmer  sold  eight  lots  of  hay,  as  follows :  7.637  tons,  3.5 
tons,  17.396  tons,  5.824  tons,  12  tons,  .95  of  a  ton,  8.0625  tons,  and 
6.4  tons.     How  much  hay  did  he  sell  ?  61.7695  tons. 

11.  In  six  consecutive  days  a  company  of  California  miners 
obtained  3.5286  ounces,  1.4  ounces,  3.125  ounces,  7.0064  ounces,  .65 
of  an  ounce,  and  2.72  ounces  of  gold.  What  was  the  whole  amount 
for  the  six  days  ?  18.43  ounces. 

12.  Capt.  Allen's  farm  consists  of  7.4  acres  of  woodland,  16.275 
acres  of  pasture,  23  acres  of  meadow,  6.025  acres  of  orchard  and 
garden,  and  72.3  acres  of  tilled  land.  How  much  land  is  there  in 
his  farm  ?  125  acres. 

13.  At  London  the  average  fall  of  rain  is,  for 

MONTHS.  INCHES.  MONTHS.      INCHES.  MONTHS.  INCHES. 

January,     1.483  May,       1.853  September,  2.193 

February,    .746  June,      1.83  October,      2.073 

March,       1.44  July,      2.516  November,  2.4 

April,         1.786  August,  1.453  December,   2.426 

What  is  the  average  annual  fall  ?  22.199  inches. 

14.  A  real-estate  agent  received  for  his  services  in  selling  seven 
farms,  $137.25,  $94.5,  $216,375,  $56.4,  $113.7,  $80,625,  and  $296.3. 
What  were  his  total  receipts  ?  $995.15. 

15.  A  grocer  bought  six  hogsheads  of  molasses,  containing  117.5 
gallons,  124  gallons,  129.3175  gallons,  104.75  gallons,  130.0625  gal- 
lons, and  131.5625  gallons.     How  much  molasses  did  he  buy  ? 

16.  What  is  the  sum  of  967  thousandths,  54  hundred-thou- 
sandths, 953  and  5  tenths,  7  and  375  thousandths,  1000  and  1  ten- 
thousandth,  6  and  75  hundredths,  8  and  80,808  hundred-thou- 
sandthSj  and  483  ?  2,460  and  40, 072  Imndred-thousandths. 


SUBTRACTION.  83 

SECTION  IV. 

STr:B  T^AC  TIOJV, 

137.  Ex.  1.  From  11.278  subtract  4.825. 
Explanation. — Since  only  like  orders  of  units  solutiok. 
in  different  numbers  can  be  subtracted  the  one  11.278 
from  the  other  (see  52,  11.),  we  write  the  num-  Jf.8 2 5 
bers  with  the  units  of  the  subtrahend,  both  deci-  6.4-63 
mal  and  integral,  under  like  orders  of  units  of 
the  minuend.  The  decimal  point  of  the  subtrahend  then 
stands  directly  under  that  of  the  minuend.  We  subtract  as 
in  integers.  Since  the  difference  between  thousandths  and 
thousandths  is  thousandths,  the  difference  between  hun- 
dredths and  hundredths  is  hundredths,  and  the  difference 
between  tenths  and  tenths  is  tenths,  and  there  are  thou- 
sandths, hundredths,  and  tenths  in  the  given  numbers, 
there  must  also  be  thousandths,  hundredths,  and  tenths  in 
their  difference.  We  therefore  place  the  decimal  point  in 
the  difference  before  the  4,  and  directly  under  the  decimal 
points  in  the  given  numbers. 

Ex.  2.  From  52  subtract  9.785. 

Explanation. — Since  decimal  ciphers  may  be  soltjtion. 

annexed  to  a  number  without  changing  its  value  5  2 .0  0  0 

(see  133,  VI.),  we  annex  ciphers  to  the  minuend  ^-  '^ ^ ^ 

until  it  has  as  many  decimal  figures  as  the  sub-  Jf.2 .2  15 
trahend,  and  then  subtract  and  place  the  deci- 
mal point  as  in  the  Solution  of  Ex.  1. 

138.   ^ule  for  Subtraction  of  decimals, 

I.  Write  the  numbers  loitli  the  decimal  point  of  the  suUra- 
hend  directly  under  that  of  the  minuend. 

II.  Subtract  in  the  same  manner  as  in  integers,  and 
place  the  decimal  point  in  the  remainder  directly  under  the 
decimal  point  in  the  subtrahend,     see  Manual, 


/ 


84  DECIMALS. 

PROBLEMS, 

(1)  (2)  (3)  (4) 

From  .3758  386.25  57.4628  2940 

Subtract      il^  50.7682  41.93  .0492 

5.  A  merchant  sold  19.25  yards  of  sheeting  from  a  piece  which 
contained  43.75  yards.     How  many  yards  were  left  ?  ^^.5. 

6.  A  company  that  contracted  to  build  a  turnpike  17.5  miles 
long,  have  completed  9.875  miles.  How  much  have  they  yet  to 
build  ?  'y^QoS  miles, 

7.  One  year  a  stock  farmer  put  43.5  tons  of  hay  into  his  bams, 
and  the  following  spring  he  had  only  8.75  tons.  How  much  had 
he  fed  to  his  stock  ?  5^.75  tons. 

8.  A  seedsman  having  73.625  bushels  of  choice  potatoes,  bought 
enough  more  to  increase  his  stock  to  120  bushels.  How  many 
potatoes  did  he  buy  ?  Jf6.375  dusMs. 

9.  The  owner  of  a  schooner  sold  .3125  of  her  to  the  captain. 
What  part  of  the  vessel  did  he  still  own  ?  .6875. 

10.  There  are  192.8125  barrels  of  water  in  a  cistern  which  will 
hold  320.5  barrels.     How  much  more  water  will  the  cistern  hold  ? 

11.  A  man  bought  a  horse  for  $118,375,  and  afterward  sold  him 
for  $130.25.     What  was  his  gain  ?  $11,875. 

12.  A  load  of  hay  with  the  wagon  weighed  2  and  65  thou- 
sandths tons,  and  the  wagon  weighed  1  and  9  hundredths  tons. 
What  was  the  weight  of  the  hay  ?  975  thousandths  of  a  ton. 

13.  A  woman  sold  a  house  and  lot,  which  cost  her  $2250.5,  for 
$1900.75.     How  much  did  she  lose  on  it  ?  $349.75. 

14.  A  person  traveled  1,200  miles  in  4  weeks,  going  276.5  miles 
the  first  week,  318.37  miles  the  second,  and  294.2  miles  the  third. 
How  far  did  he  travel  the  last  week?  310.93  miles. 

15.  A  vessel  of  400  tons  burthen,  bound  up  the  lakes,  ships  at 
Buffalo  93.4  tons  of  railroad  iron,  56.81  tons  of  salt,  and  211.7  tons 
of  general  merchandise.     How  much  does  she  lack  of  a  full  cargo  ? 

10.  A  man  having  three  farms,  containing,  respectively,  296,5 
acres,  145.75  acres,  and  96  acres,  sold  to  one  man  72.5  acres,  and  to 
another  86  acres,  and  gave  to  each  of  his  two  sons  105.25  acres. 
How  many  acres  had  he  left  ?  169.25. 


MULTIPLICATION. 


85 


SECTION   V. 

139.  2  times  4  are  8,  3  times  3  are  9,  5  times  7  are  35  (or 
35  ones)  =  3  tens  and  5  ones.  In  each  of  these  illustra- 
tions the  two  factors  are  ones  and  the  product  is  ones. 

The  product  of  ones  multiplied  ly  ones  is  alimys  ones. 

140.  In  multiplying  24.3  by  2,  the  8  of  the  prod- 
uct is  obtained  by  multiplying  the  4  ones  of  the 
multipKcand  by  the  2  ones  of  the  multiplier. 
Hence,  the  8  is  ones,  and  the  decimal  point  must 
be  placed  at  the  right  of  it. 

In  multiplying  4.17  by  2.1,  the 
product  of  the  4  ones  of  the  multi- 
plicand and  the  2  ones  of  the  multi- 
pher  is  the  8  of  the  second  partial 
product.  Hence,  that  figure  is  ones, 
and  the  8  of  the  final  product  is  also 
ones. 

In  Ex.  3,  the  8  of  the  third  partial  product  is  the  product 
of  the  ones  of  the  multiplicand  and  the  ones  of  the  multi- 
plier. Hence  it  is  ones,  and  the  9  of  the  final  product  is 
also  ones- 

In  Ex.  4,  the  8  in  the  sec- 
ond partial  product  is  ones, 
and  the  9  in  the  final  product 
is  also  ones  ;  and  in  Ex.  5,  the 
7  of  the  third  partial  product 
is  ones,  and  also  the  8  of  the 
final  product. 

141.  In  Ex.  1  there  is  one  decimal  figure  in  one  of  the 
factors,  and  one  decimal  figure  in  the  product.  In  Ex.  2 
there  are  three  decimal  figures  in  the  factors  and  three  in 
the  product.    In  Ex.  3  and  4  there  are  four  decimal  figures 


Ex  2. 

J,. 17 

2J 

8.3  J^ 

8,75  7 


Ex.  1. 


Jf8.6 

Ex.8. 

J^.26 
2.13 

127S 
U2G 
8.52 

9.07  3  8 


Ex.4. 

Jf.316 
32.U 

1726  Jf. 
8.6  3  2 
1291^8 

139.8381/. 


Ex.  5. 

2,501/3 
3.1/2 

50086 
100172 
7.5  12  9 

8.561/706 


86 


DECIMALS. 


is  ten-tliou- 
sandths : 


1    u 

\6\3 
2\A\0 

2 
0 
3 

0  \  6 
9  1 

2  L? 

1  ^1 

1  T 

5 

1 

9\  6 

•9    1    1 

1  ll 
III 

in  tlie  two  factors  and  four  in  the  product ;  and  in  Ex.  5 
there  are  six  decimal  figures  in  the  factors  and  six  in  the 
product.     From  these  examples  we  learn  that 

The  member  of  decimal  figures  in  a  product  must  equal  tlie 
number  of  decimal  figures  in  its  factors. 

1 42.  By  examining  this  Example  we  see  Ex. 

that  the  2  of  the  third  partial  product  is  2.103 

the  product  of  ones  multiplied  by  ones,  1.32 
and  therefore  must  be  ones  ;  and  that 

'  of  ones  and  tenths  is  tenths ; 
of  ones  and  hundredths,  )  .^  i,^j,^iredths 
and  of  tenths  and  tenths  ' 
of  ones  and  thousandths,  [  is  thou- 

The        and  of  tenths  and  hundredths  )  sandths 
product  "j  of  tenths  and  thousandths, 
and  of  hundredths  and  hun 

dredths 
of  hundredths  and  thousandths  is  hun- 
dred-thousandths.   Hence, 

Tlie  number  of  tlie  decimal  place  in  which  the  product  of 
any  tivo  decimal  figures  belongs,  counting  from  ones,  is  equal 
to  the  sum  of  the  numbers  of  the  decimal  places  of  the  two 
figures  multiplied. 

113*  Since  3  times  0  ones  is  0  ones,  we 
place  the  decimal  point  in  Ex.  1  at  the 
right  of  the  0  in  the  product,  as  shown 
in  (1).  But  the  0  may  be  omitted  from 
the  multiplicand  without  changing  its 
value,  and  the  product  will  then  be  .759, 
as  shov/n  in  (2). 

In  multiplying  4  ten-thousandths  by  2 
thousandths,  since  the  4  is  four  places  and 
the  2  is  three  places  to  the  right  of  ones, 
the  product,  8,  must  be  seven  places  to  the 
right  of  ones,  and  the  other  six  places  must 
be  filled  by  ciphers.     Hence, 


Ex.1. 
0) 
0.253 
£ 

0.759 


(2) 

.253 
5 

.759 


Ex.2. 

.OOOJf. 
.002 

0000008 


^  m  TIT' 
IVEi 

MULTIPLICATION.      X^T^  *•  T'?^?7 

When  there  are  not  as  many  figures  i7i  the  product  as  tKere 
are  decimal  figures  in  the  factors,  decimal  ciphers  must  le 
prefixed  to  the  other  figures  to  supply  the  deficiency. 

14li  Upon  the  principles  deduced  in  Art  141,  142,  143, 

is  based  the 

^tele  for  MuUipUcation  of  decimals. 

I.  Write  the  numters  and  multiply  as  in  integers. 

II.  Place  the  decimal  point  in  the  product,  so  that  it  shall 
contain  as  many  decimal  figures  as  hoth  factors. 

I'M  OBIjEMS. 

1.  How  many  gallons  of  oil  are  there  in  16  barrels,  each  contain- 
ing 31.5  gallons  ?  50^. 

2.  In  a  certain  manufactory  3.7  tons  of  coal  are  used  each  day. 
How  much  will  be  used  in  26  days  ?  *96.2  tons. 

3.  A  farmer  sows  1.75  bushels  of  wheat  to  the  acre.  How  much 
seed  will  he  require  to  sow  19  acres  ?  33.25  lusJiels. 

(4)  (5)  (6)  (7) 

33.125  miles  14.6  27.31  24753 

27  _3^  4.5  3.16 

8.  Bought  137.5  acres  of  land,  at  $76.25  per  acre.  What  did  the 
whole  cost  ?  $10484.375. 

9.  If  62.5  tons  of  iron  be  required  for  the  track  of  one  mile  of 
railroad,  how  much  iron  will  it  take  for  371.75  miles? 

10.  One  pound  English  money  is  worth  $4.84  United  States 
money.  What  is  the  value  in  United  States  money  of  16.87  pounds 
English  money  ?  $81.6508. 

11.  What  was  the  length  of  an  army  wagon- train  that  passed  a 
given  point  in  4.08  hours,  passing  at  the  rate  of  3.025  miles  per 
hour  ?  12.342  miles. 

12.  A  mowing-machine  company  bought  137.5  tons  of  iron,  at 
$32.75  a  ton.     To  what  did  the  purchase  amount  ?       $4503.125. 

13.  Brass  is  .8  copper  and  .2  zinc.  How  much  copper  and  how 
much  zinc  must  be  used  to  make  .875  of  a  ton  of  brass  ? 

Copper,  .7  of  a  ton;  zinc,  .175  of  a  ton. 


88  DECIMALS. 

14.  A  man  owning  .8  of  a  mill,  sold  .3125  of  his  share.  What 
part  of  the  mill  did  he  sell  ?  .25  of  it. 

15.  If  the  length  of  a  military  step  is  2.25  feet,  how  far  will  a 
soldier  march  in  taking  1,762  steps  ?  3964.5  feet. 

IG.  One  week  a  butcher  bought  354  lambs,  at  $4.7  per  head. 
How  much  did  they  cost  him  ? 

17.  If  the  average  rate  of  speed  of  a  railroad  freight  train,  in- 
cluding stops,  is  11.88  miles  per  hour,  how  far  will  it  run  in  .85  of 
an  hour  ? 

18.  Limestone  loses  .35  of  its  weight  when  weighed  in  water.  If 
a  piece  of  limestone  weighs  17.137  ounces  in  air,  how  much  less  will 
it  weigh  in  water  ?  5.99795  ounces. 

19.  A  cubic  inch  of  silver  weighs  6.0G13  ounces,  and  gold  weighs 
1.83865  times  as  much  as  silver.  What  is  the  weight  of  a  cubic 
inch  of  gold  ?  ll.lU6092Jt5  ounces. 

(20)  (21)  (22)  (23) 

72.65         .92         .000873         .00096 

.6         ^  .26         .01298 

24.  What  is  the  product  of  .0625  and  .48  ?  .03. 

25.  What  is  the  weight  of  25.75  feet  of  copper  pipe,  if  one  foot 
Weighs  .375  of  a  pound  ?  8.65625  ^pounds. 

26.  If  one  man  can  mow  1.875  acres  in  a  day,  how  many  acres 
can  13  men  mow  in  7.5  days  ?  182.8125. 

27.  I  made  325  gallons  of  cider.  How  much  had  I  left,  after 
selling  9  barrels,  each  containing  31.5  gallons  ?  Ji^l.5  gallons. 

28.  On  invoicing  his  stock,  a  merchant  finds  that  he  has  7  pieces 
of  cotton  goods,  of  43.75  yards  each,  4  j)ieces  of  46.5  yards  each, 
3  pieces  of  39.5  yards  each,  one  piece  of  24.375  yards,  and  one  of 
19.675  yards.     How  many  yards  has  he  in  all  ?  65Jf.8. 

29.  A  farmer  sowed  three  fields  to  rye.  The  first,  of  13.5  acres, 
yielded  23  bushels  per  acre ;  the  second,  of  9  acres,  yielded  30.25 
bushels  per  acre  ;  and  the  third,  of  11.75  acres,  yielded  24.44  bush- 
els per  acre.      What  was  the  total  yield  ?  869.92  Imshels. 

30.  How  many  bushels  of  oats  must  a  livery  stable  keeper  buy 
to  last  11  horses  19  weeks,  if  he  feeds  to  each  horse  2.625  bushels  a 
week  ?  548.625. 


DIVISION.  89 

SECTION  VI. 
^Dirisiojy. 

145.  We  have  learned,  in  Art.  95,  that  in  the  division  of 
integers  any  quotient  figure  must  be  of  the  same  order  of 
units  as  the  right-hand  figure  of  that  part  of  the  dividend 
used  to  obtain  it. 

If  we  divide  6  tenths,  or  .6,  by  3,  (i)  (2) 

the  quotient  is  2  tenths,  or  .2.     If        :^  I  ^         ^  [  5 
we  divide  6  hundredths,  or  .06,  by        ,2  ,02 

3,  the  quotient  is  2  hundredths,  or 
.02.      The    quotient    of    6     thou- 
sandths, or  .006,  divided  by  3,  is  ^^\  ^^^ 
2   thousandths,    or   .002,   and  the     'l^V^       .0006  [3 
quotient  of  .0006,  divided  by  3,  is    -002            ,0  002 
.0002.     In  other  words, 

When  tenths       are  divided  by  an  integer,  the  quotient  is  tenths ; 

"     hundredths         "  "         "  "  hundredths; 

"     thousandths        "  "         "  "  thousandths; 

"     ten-thousandths"  "         "  "  ten-thousandths; 

and  so  on. 

When  the  divisor  is  an  integer y  any  quotient  figure  will  be  of 
the  same  order  of  units,  integral  or  decimal,  as  the  right-hand 
figure  of  the  partial  dividend  used  to  obtain  it. 

146.  Ex.  Divide  16.285  by  5. 

Explanation. — We  write  the  terms,  and  solution. 

commence  at  the  left  to  divide,  as  in  in-       IG .28 5  {5 
tegers.     Since  the  first  partial  dividend,  16,  3.257 

is  ones,  the  first  quotient  figure,  3,  must  be 
ones,  and  the  next  quotient  figure  will  be  tenths.   We  there- 
fore place  the  decimal  point  after  the  3  ones,  before  writing 
any  of  the  other  figures  of  the  quotient. 

The  decimal  point  must  always  be  placed  in  the  quotient,  be- 
fore writing  the  tenths'  figure. 


90  DECIMALS. 

147.  Ex.  1.  Divide  .0056  by  4 
Explanation.  —  Since  4  is  contained  in  0        solution. 

tenths  0  times,  and  in  0  hundredths  0  times,      -0056  { ^ 
we  write  ciphers  in  the  places  of  tenths  and      ,00  IJf. 
hundredths  after  the   decimal  point  in  the 
quotient. 

Ex.  2.  Divide  .1272  by  8. 

Explanation.  —  Since  8  is   contained  in   1  solution. 

tenth  0  times,  we  write  a  cipher  in  place  of      'J-^'7 ^  {8 
tenths  after  the  decimal  point  in  the  quotient.      ,015  9 
Hence, 

When  the  first  decimal  figure  or  figures  of  the  dividend  will 
not  contain  the  divisor,  a  decimal  cipher  or  ciphers  must  be 
written  in  the  quotient. 

148.  Ex.  1.  Divide  12.6  by  24. 
Explanation.  — 12.6  -4-  24  =  .5,  with  a  re-         solution. 

mainder  of  6  tenths.  6  tenths  =  60  hun-  ^^-^  _^ 
dredths  (see  129),  and  60  hundredths  -^  24  =  —  [.525 
2  hundredths,  with  a  remainder  of  12  hun-  ^0 

dredths.     12  hundredths  =  120  thousandths  ^^ 

(see    129),    and   120    thousandths  -^  24  =  5  -?^^ 

thousandths.     Hence,  -^    ^ 

When  there  is  a  remainder  after  using  all  the  figures  of  the 
dividend,  the  division  may  ie  continued,  each  new  partial  divi- 
dend being  formed  by  annexing  a  decimal  cipher. 

Ex.  2.  Divide  31.5  by  8. 
Explanation. — ^In  this  Solution  we  form  each      solution. 
partial  dividend  after  the  second,  by  mentally    ^  ^-^  I  ^ 
annexing  a  decimal  cipher  to  the  partial  re-       3.9375 
mainder. 

JPH  OBTjEMS. 

1.  A  father  divided  280.5  acres  of  land  equally  among  3  sons. 
How  much  land  did  each  receive  ?  93.5  acres. 

3.  In  how  many  weeks  will  a  man  whose  wages  are  $9  a  week, 
earn  $157.5  ?  17.5. 


DIVISION.  91 

3.  If  a  ditcher  digs  8  rods  of  ditch  in  one  day,  how  long  will  it 
take  him  to  dig  118  rods  ?  H.75  days. 

4.  A  farmer  made  45  barrels  of  cider  from  292.5  bushels  of 
apples.    How  many  apples  did  it  take  for  a  barrel  of  cider  ? 

(5)  (6)  (7)  (8) 

209.58  1 6  $7209  [  $8  341.5  [  77  .1537  1 29 

9.  If  18  silver  spoons  weigh  33.75  ounces,  what  is  the  weight  of 
1  spoon  ?  1,875  ounces. 

10.  If  one  sheet  of  paper  makes  48  pages  of  a  book,  how  many 
sheets  will  it  take  for  a  book  of  348  pages  ?  7.25. 

11.  A  tailor  cut  6  coats  from  13.75  yards  of  broadcloth.  How 
much  cloth  did  he  put  into  a  coat  ? 

149.  2  is  contained  in  6,  3  times  ;  2  tens  in  6  tens,  3 
times ;  2  Imndreds  in  6  hundreds,  3  times ;  and  so  on. 
So,  also,  2  tenths,  or  .2,  is  contained  in  6  tenths,  or  .6,  3 
times ;  2  hundredths,  or  .02,  is  contained  in  6  hundredths, 
or  .06,  3  times  ;  2  thousandths,  or  .002,  is  contained  in  6 
thousandths,  or  .006,  3  times  ;  and  so  on.     That  is. 

When  the  divisor  and  dividend  are  of  the  same  order  of 
units,  either  integral  or  decimal^  the  quotient  is  ones. 

This  truth  is  shown  in  the  following  examples  : 

(Ex.  1)  (Ex.  2)      (Ex.  8)      (Ex.  4)        (Ex.  5)  (Ex.  6) 

6001200        60{20        6  12        .6  [.2         .06^{.02        .006  [ .002 
3  3  8  3  3  3 

150.  Ex.  1.  Divide  36.45  by  .15.  solution. 

Explanation. — Since  the  right-hand  figure 
of  both  divisor  and  dividend  is  of  the  same 
order  of  units  (hundredths),  the  right-hand 
figure  of  the  quotient  must  be  ones,  and 
consequently  the  entire  quotient  is  an  in-  ^5 

teger.  ^tz^ 


S6.Jf5 

.15 

SO 

2Jf3 

6J, 

60 

92  DECIMALS. 

Ex.  2.  Divide  2.68  by  .25. 
Explanation.  — The  right-    partial  solution. 
hand  figure   of   both  divi-     ^-^^   I -^^ 

sor  and  dividend  being  hun-     I  -^  ^ 

dredths,  the  10  of  the  quo-        ^  ^ 

tient  is  an  integer,  as  shown 

in  the  Partial  Solution.    But  since  there 

is  a  remainder,  we  place  a  decimal  point 

after  the  10,  and  continue  the  division  by 

annexing  decimal  ciphers  to  the  partial  remainders.  (See  148). 

Ex.  3.  Divide  15.695  by  7.3. 

Explanation.  —  The  quotient  of  15.6 
(the  first  three  figures  of  the  dividend) 
divided  by  7.3,  is  an  integer,  because  the 
right-hand  figure  of  each  term  is  tenths. 
We  therefore  place  an  inverted  caret  (V) 
after  the  6  of  the  dividend,  to  show  what 
figures  are  used  to  obtain  the  integral 
part  of  the  quotient.  Placing  the  decimal 
point  after  the  quotient  figure,  2,  we  complete  the  division 
and  obtain  the  entire  quotient,  2.15. 

The  number  of  decimal  figures  in  the  quotient  will  equal  the 

number  of  decimal  figures  left  in  the  dividend^  after  taking 

from  it  as  many  decimal  figures  as  there  are  decimal  figures 

in  the  divisor. 

Note.— When  decimal  ciphers  are  annexed  to  form  partial  dividends, 
they  must  be  counted  as  decimal  figures  of  the  dividend. 

151.  Ex.  Divide  42  by  .56. 

Explanation.— The  right-hand  figure  of  the  soltttion. 

divisor  is  hundredths  ;  and  as  the  right-hand  |^-^^  j  "^ 

figure  of  the  dividend  must  also  be  hundredths     !_.  [  7  5 

to  obtain  ones  for  the  quotient   (see  149),  we  ^^0 

annex  two  decimal  ciphers  to  the  dividend  be-       Z. 

fore  dividing. 


15.6' 9  5 

7.3 

lJf-6 

2.15 

109 

78 

365 

865 

DIVISION.  93 

The  dividend  must  contain  at  least  as  many  decimal  figures 
as  the  divisor. 

152.  Upon  the  principles  deduced  in  Art.  145,  147,  148, 
150,  is  based  the 

^iite  for  division  of  decimals. 
I.  When  the  divisor  is  an  integer. 

1.  Jf  necessary,  annex  decimal  ciphers  to  the  dividend,  till  the 
figures  of  the  dividend  will  contain  the  divisor. 

2.  Divide  as  in  integers. 

3.  Place  the  decimal  point  in  the  quotient  so  that  it  shall  con- 
tain as  many  decimal  figures  as  the  dividend. 

II.  When  the  divisor  is  a  decimal  or  a  mixed  number. 

1.  Place  an  inverted  caret  after  the  figure  of  the  dividend 
that  is  of  the  same  order  of  units  as  the  right-hand  figure  of 
the  divisor. 

2.  Divide  as  in  integers,  and  place  the  decimal  point  so  that 
the  quotient  shall  contain  as  many  decimal  figures  as  there  are 
decimal  figures  at  the  right  of  the  inverted  caret  in  the  dividend. 

PM  OBLEMS. 

12.  How  many  dress  patterns,  of  11.5  yards  each,  are  there  in  a 
piece  containing  46  yards  of  delaine  ?  4- 

13.  If  one  length  of  6-inch  stove  pipe  can  be  made  from  3.14 
pounds  of  Russia  iron,  how  many  lengths  can  be  made  from  72.23 
pounds  ?  23. 

(14)  (15)  (16)  (17) 

75.6  [.9  21.25  [.4  99^24.75  3.985  [159.4 

18.  How  many  casks,  each  holding  41.315  gallons,  will  be  re- 
quired to  hold  11278.995  gallons  of  alcohol  ?  273. 

19.  A  merchant  exchanged  35.0625  yards  of  cloth  for  wood,  at 
the  rate  of  4.125  yards  for  1  cord.  How  much  wood  did  he  re- 
ceive ?  8.5  cords. 

20.  A  miller  received  $3,009  for  ship-stuflfs,  at  $21.25  per  ton. 
How  many  tons  did  he  sell  ?  m.6. 


94  DECIMALS. 

21.  How  long  will  it  take  to  manufacture  1321.65  barrels  of  flour, 
at  the  rate  of  53.4  barrels  per  day  ?  2Jf.75  days. 

22.  At  Catskill,  N.Y.,  on  the  26th  of  July,  1860,  the  extraordi- 
nary fall  of  18  inches  of  rain  occurred  in  7.5  hours.  What  was  the 
average  fall  per  hour  ?  S.J/,  inches. 

23.  If  7  men  cradle  116.55  acres  of  grain  in  4.5  days,  how  many 
acres  does  1  man  cradle  in  1  day  ?      See  Manual.  3.7. 

24.  The  winter  term  of  a  country  school  continued  13  weeks  of 
5.5  days  each,  and  the  aggregate  attendance  for  the  whole  term 
was  3074.5  days.    What  was  the  average  daily  attendance  ?     Ji3. 

153t  Ex.  How  many  goblets,  each  weighing  7.5  ounces, 
can  a  manufacturer  make  from  176  ounces  of  silver  ? 

Explanation. — Since  he  will  not  make  the  solution. 

decimal  part  of  a  goblet,  the  result  in  this  -^'^^  J^ 
problem  will  be  an  integer;  and  the  solution  is  .  ,  [  2  3 
complete  when  the  ones  of  the  quotient  are  ^^0 

obtained.     Since  the  260  is  tenths,  the  35  is  ?H 

tenths,  and  the  true  remainder  is  3.5.    Hence,  ^'^ 

The  right-hand  figu7'e  of  any  remainder  will  ahvays  ie  of 
the  same  order  of  units,  integral  or  decimal^  as  the  last  figure 
of  the  dividend  used  to  oUadn  it. 

PMOBLEMS. 

25.  Into  how  many  building  lots,  each  containing  .375  of  an  acre, 
can  5  acres  of  land  be  divided  ?  13^  with  .125  of  an  acre  left. 

26.  An  oil  refiner  has  on  hand  22,240  gallons  of  oil.  How  many 
casks  can  he  fill,  if  he  puts  42.5  gallons  in  each  cask  ? 

523 f  and  have  12.5  gallons  left. 

27.  A  forwarder  has  2,150  tons  of  freight  to  ship  by  canal.  If 
110.5  tons  make  one  boat-load,  how  many  boat-loads  has  he  ? 

50.5  tons  more  than  19  'boat-loads. 

28.  If  a  teamster  draws  1.125  cords  of  wood  at  a  load,  how  many 
loads  will  41.75  cords  make  ?  37 ^  and  .125  of  a  cord  more. 

29.  How  many  potash  kettles,  each  weighing  362.5  pounds,  can 
be  made  from  20,500  pounds  of  iron  ?         Remainder^  200  pounds. 


UNITED  STATES  MONEY. 


95 


154.  United  States  Money,  or  JF'ederat  J^foneyy  con- 
sists of  dollars,  cents,  and  mills. 


10  mills  are  1  cent. 
100  cents  are  1  dollar. 


1  dollar  is  100  cents. 
1  cent     is    10  mills. 


The  unit  of  United  States  Money  is  the  dollar. 

155.  Since  100  cents  are  1  dollar,  1  cent  is  1  hundredth 
of  a  dollar ;  and  since  10  mills  are  1  cent,  1  mill  is  1  tenth 
of  a  cent,  or  1  thousandth  of  a  dollar.     Hence, 

Cents  may  always  he  written  as  hundredths,  and  mills  as 
thousandths  of  a  dollar. 

1  cent    is  written  $.01. 
47  cents  are  written  $.Ji7. 

1  mill    is  written  $.001. 
50  cents  4  mills  are  written  $.504. 
25  dollars  5  cents  8  mills  are  written  $25,058. 
100  dollars  5  mills  are  written  $100,005. 


96  DECIMALS. 

Note. — The  Table  of  United  States  Money,  as  established  by  Act  of  Con- 
gress, August  8, 1786,  is  as  follows :' 

10  mills   are  1  cent ; 
10  cents    "    1  dime ; 
10  dimes   "   1  dollar ; 
10  dollars  "    1  eagle. 
But  dimes  are  always  read  as  tens  of  cents,  and  eagles  as  tens  of  dollars. 
Thus,  7  eagles  2  dollars  4  dimes  5  cents  is  $73.45,  and  is  read  "  72  dollars  45 
cents,"  or  "  72  and  45  hundredths  dollars." 

JEXEMCISES, 

1.  Read  $.06,  $.44,  $.80,  $3.15,  $70.40,  $9.08. 

2.  Read  $.005,  $.456,  $.047,  $.192,  $.601,  $.309. 

3.  Read  $19,476,  $500,104,  $1,008,  $297,027. 

4.  Write  20  cents ;  5  cents ;  93  cents. 

5.  Write  10  dollars  50  cents ;  150  dollars  88  cents. 

6.  Write  4  mills ;  26  cents  9  mills ;  5  cents  3  mills. 

7.  Write  5  dollars  17  cents  5  mills. 

8.  Write  200  dollars  4  cents  8  mills. 

9.  Write  30  dollars  6  mills. 

156.  Decimal  parts  of  a  dollar  less  than  mills  or  tTiou- 
sandths  are  read  as  decimals  of  a  mill. 

$.0005  is  5  tenths  of  a  mill. 

$.00025  is  25  hundredths  of  a  mill. 

$.0064  is  6  and  4  tenths  mills. 

$.3765  is  37  cents  6  and  5  tenths  mills. 

$45.40375  is  45  dollars  40  cents  3  and  75  hundredths  mills. 

EXEJt  CISES. 

10.  Read  $.0004,  $.0056,  $.00075,  $.3715. 

11.  Read  $.47675,  $93.7564,  $300.85354. 

12.  Write  5  tenths  of  a  mill ;  75  hundredths  of  a  mill. 

13.  Write  5  and  8  tenths  mills ;  4  cents  2  and  9  tenths  mills. 

14.  Write  56  cents  4  and  72  hundredths  mills. 

15.  Write  8  dollars  10  cents  1  and  38  hundredths  mills. 

157.  A  Coin  is  a  piece  of  metal  on  which  certain  char- 
acters are  stamped,  by  authority  of  the  General  Govern- 
ment, making  it  legally  current  as  money. 


UNITED    STATES    MONEY. 


97 


United  States  coins  arc  made  of  gold,  silver,  nickel,  and 
copper,  as  shown  in  the 


COIN    TABLE. 


Gold, 


NAMES  OF  COINS. 

VALUES. 

METALS. 

NAMES  OF  COINS. 

VALUES. 

50-dollar  piece, 

$50.00 

[Dollar, 

$1.00 

Double  eagle, 

30.00 

Half-dollar, 

.50 

Eagle, 

10.00 

Silver,   - 

Quarter-dollar, 

.25 

Half-eagle, 

5.00 

Dime, 

.10 

3-dollar  piece, 

3.00 

Half-dime, 

.05 

Quarter-eagle 

3.50 

,  3-cent  piece, 

.03 

Dollar, 

1.00 

5-cent  piece, 

.05 

2-cent  piece. 

.03 

Nickel, . 

3-cent  piece. 

.03 

Cent, 

.01 

.  Cent, 

.01 

Notes.— 1.  Gold  coins  of  the  values  of  $.50  and  $.25  were  coined  by  pri- 
vate assayers  in  California,  the  former  in  the  years  1853-53,  and  the  latter 
in  1854. 

3.  Half-cent  copper  coins  have  not  been  coined  since  the  first  issue  of  the 
nickel  cent  in  the  year  1857. 

3.  3-cent  pieces  of  copper  and  nickel  -were  first  issued  in  the  year  1865, 
and  5-cent  pieces  of  the  same  metals  in  1866. 

4.  The  50-dollar  piece  shown  in  the  cut,  page  95,  is  about  .8  as  large 
across  as  the  real  coin ;  the  other  coins  shown  in  the  cut  are  full  size. 

See  Manual. 

158.  AI^OJ  is  a  baser  metal  mixed  with  a  finer ;  as  silver 
with  gold,  or  copper  with  silver. 

159.  The  United  States  gold  and  silver  coins  consist  of  9 
parts  or  .9  by  weight  of  pnre  metal,  and  1  part  or  .1  of 
alloy  ;  the  alloy  of  gold  coins  being  equal  parts  by  weight 
of  silver  and  copper,  and  that  of  silver  coins  pure  copper. 

Nickel  and  copper  coins  are  not  alloyed.  see  Manual. 

160.  In  final  results  of  computations,  and  in  business 
transactions 


$.005    are 

written 

$.00^, 

and 

read  one  half   cent. 

$.0035 

(I 

$.00i 

one  fourth         " 

$.0075 

(( 

$.00f. 

three  fourths   " 

$.00135 

u 

$.00i. 

one  eighth       " 

$.00375 

u 

$.00|, 

three  eighths   " 

$.00635 

u 

$.00|, 

five  eighths      " 

$.00875 

u 

$.001, 

seven  eighths  " 

See  Manual. 

98 


DECIMALS, 


COMIFXJT^TIONS  OJB^  xj.  s,  m:o:ney. 

161.  Since  the  dollar  is  the  unit  of  United  States  Money, 
(see  154),  and  cents,  mills,  and  parts  of  a  miU  are  decimals 
of  a  dollar,  it  follows  that 

United  States  Money  is  added,  sultracted,  multiplied,  and 
divided  in  the  same  manner  as  other  decimals. 

SOLUTION. 

162.  Ex.  1.  What  is  the  sum  of  $275,10,       $27  5.10 

$18.37i,  $.883,  and  $31  ?  18.3  7  5 

Explanation. — "We  write  the  numbers  with  .8  8  3 

3  1 
dollars  under  dollars,   cents  under  cents, 

and  mills  under  mills ;  and  then  add  the 

parts,  and  place  the  decimal  point  in  the 

sum,  as  in  Addition  of  Decimals. 


Ex.  2.  From  $52.75  subtract  $10.96|. 

Explanation.  —  "We  write  dollars  under 
doUars,  and  cents  under  cents ;  and  then 
subtract,  and  place  the  decimal  point  in  the 
remainder,  as  in  Subtraction  of  Decimals. 

Ex.  3.  Multiply  $45,625  by  5.6, 
Explanation.  —  "We  write  the  multiplier 
under  the  multipHcand  ;  and  then  multiply, 
and  place  the  decimal  point  in  the  product, 
as  in  Multiplication  of  Decimals,  Omitting 
two  decimal  ciphers  from  the  right 
of  the  product,  (see  133,  VI),  the 
required  product  is  $255.50. 


$325,358 

SOLUTION. 

$5  2.7  5 
10.9675 

$Ji.l.7825 


SOLUTION. 

$1^5.625 
5.6 

273750 
228125 

$255.5000 


$lJfJt.5. 
82 


25 


8. 


$17,625 


Ex.  4.  Divide  $1445.25  by  82. 

Explanation.  —  "We  write  the 
divisor  at  the  right  of  the  divi- 
dend; and  then  divide,  and  place 
the  decimal  point  in  the  quo- 
tient, as  in  Division  of  Deci- 
mals. 


625 
5_7Jt_ 

512 

Jf.92 


205 
16J_ 

UIO 
It-IQ 


UNITED    STATES    MONEY.  99 

Ex.  5.   How  many  times  are  solution. 

$5.06i  contained  in  $567 ?  $567.000  0  ^.  $5.0625 

Explanation. — We  divide  as  in        50625        |     ^  ^ ^ 
Division  of  Decimals.  Since  both  60750 

dividend   and  divisor   are   con-  50625 

Crete  numbers  (dollars),  the  quo-  101250 

tient  must  be  an  abstract  num-  101250 

ber.     (See  97). 

163i  In  business  transactions,  when  the  mills  in  any  fined 
result  are  5  or  more,  they  are  regarded  as  1  cent ;  and  when 
less  than  5,  they  are  rejected. 

164i  The  commercial  character  @  signifies  at,  or  by  the 
yard,  pound,  gallon,  bushel,  or  other  unit  named  in  the 
problem.  Thus,  "2  dozen  eggs,  ®  $.28,"  signifies  ''2  dozen 
eggs,  at  $.28  a  dozen." 

Pit  OBLEMS. 

1.  Martha  paid  $.87|-  for  a  grammar,  $.25  for  a  slate,  $.75  for  a 
reader,  and  $.12^  for  a  writing-book.  What  was  the  amount  of 
her  purchases  ?  $2, 

2.  A  farmer  killed  an  ox,  and  sold  the  four  quarters  for  $9,935, 
$9.62^,  $8.11,  and  $8 ;  the  hide  for  $6.89 ;  and  the  tallow  for  $8.92. 
How  much  did  the  ox  bring  him  ?  $51.48. 

3.  One  year  a  gentleman's  income  tax  was  $34.26 ;  his  state  tax 
was  $42.11;  village  tax,  $18.04 ;  school  tax,  $7.65 ;  road  tax,  $.62 J- ; 
and  military  tax,  $1.  What  amount  of  taxes  did  he  pay  that 
year  ?  $103.68^. 

4.  A  owes  to  B,  $374 ;  to  C,  $47.50 ;  to  D,  $193.1875 ;  to  E, 
$21.81 ;  to  F,  $6.75  ;  to  G,  $3,125  ;  and  to  H,  $11.0625.  What  is 
the  amount  of  his  indebtedness  ?  $657,435, 

What  is  the  amount  of  each  of  the  following  bills  ? 

7.   Traveling  Expenses. 
Railroad  fare,  $18,625 
Steamboat  "       7.25 
Carriage  hire,      5.00 
Hotel  bills,     .  31.875 
Other  expenses,  17. 6 7 


5.  For  Furniture. 

6.  For  Hardware. 

1  Set  of  Chairs,  $7.50 

1  Plow,  .     .  $6.75 

1  Table,  .     .     .    4.75 

1  Spade,     .     1.125 

1  Eocking-chair,  3.25 

1  Hammer, .      .5625 

1  Bedstead, .     .    7.25 

1  Pitchfork,      .875 

1  Mirror,      .    .    1.375 

Nails,     .     .     1.375 

100  DECIMALS. 

8.  A  man  having  $300.82  in  bank,  draws  out  $00.29.  What  ia 
then  his  balance  in  bank  ?  $2Jf0.53, 

(9)  (10)  (11)  (13) 

From  $593,025         $1132.053        $251  $1574 

take      490.54  90  8.375  .856 

13.  A  man  gave  me  his  note  for  $75,  and  he  has  since  paid  all 
but  $24.50  of  it.     How  much  has  he  paid  on  the  note  ?      $50.50. 

14.  A  dress-maker  earned  $34  in  a  month,  and  her  expenses  were 
$26.67.    How  much  did  she  save  ?  $7.33. 

15.  A  grain  buyer  purchased  a  lot  of  wheat  for  $1078.25,  and  the 
following  week  sold  it  for  $1219.125.  How  much  did  he  clear  on 
the  wheat  ?  $140.87^-. 

10.  I  paid  $2841.375  for  an  interest  in  an  iron-foundery,  and 
afterward  sold  it  for  $3129.16,     How  much  was  my  gain  ? 

(li)  (18)  (19)  (20) 

Multiply  $21.25  $2.4375  $0.80  $4,025 

by  46  .215  12.5  11.25 

21.  How  much  will  24  bushels  of  turnips  come  to,  at  $.375  a 
bushel  ?  $9. 

22.  A  lady  bought  32  yards  of  carpeting,  @,  $1,121^.  How  much 
did  it  cost  her  ?  $36. 

23.  How  much  must  I  pay  for  23  rolls  of  paper-hangings, 
©  $.375  ?  $8.6^. 

24.  A  farmer  sold  390  pounds  of  wool,  @  $.075.  How  much  did 
it  coiue  to  ?  $267.30. 

25.  A  mechanic  worked  4.9  days,  for  $1,875  per  day.  How 
much  did  he  earn  ?  $9.18^. 

20.  If  the  interest  on  $1  for  1  year  is  $.07,  what  is  the  interest  on 
$24.75  ?  $1.73^. 

27.  A  drover  bought  a  flock  of  123  sheep,  ®  $2.5025.  What 
was  the  cost  of  the  flock  ?  $315.18}. 

28.  A  milk-man's  sales  average  219  quarts  a  day,  at  $.05  a  quart. 
What  are  his  daily  receipts  ? 

29.  How  much  do  his  sales  amount  to  in  a  year,  or  305  days  ? 

$3996.75. 


UNITED    STATES    MONEY.  101 

30.  I  paid  $19.9375  for  iron  castings,  at  ^.0625  a  pound.     How 
many  pounds  of  castings  did  I  buy  ?  319. 

(31)  (32)  (33)  (34) 

$5.25  [14        $362.25  [23        $180.40  [$.1025        $63  [56 


85.  A  shipment  of  1,583  bushels  of  corn  was  sold  for 
What  was  the  price  per  bushel  ?  ^.56f . 

36.  A  tanner  paid  $156.82|-  for  25.5  cords  of  hemlock  bark. 
How  much  was  that  a  cord  ?  $6.15. 

37.  Divide  $6  into  150  equal  parts. 

38.  How  many  bushels  of  potatoes  can  be  bought  for  $57.3125, 
at  $.875  per  bushel?  65.5. 

39.  A  builder  contracted  to  put  up  a  brick  dwelling  for  $3,725. 
Tlie  building  materials  cost  him  $1641.0625,  and  he  paid  out  for 
labor  $1296.50.  Did  he  make  or  lose  money  on  the  contract,  and 
how  much  ?  Be  made  $787.43§. 

40.  A  man  exchanged  a  horse  worth  $187.50,  and  a  watch  worth 
$64,875,  for  a  span  of  horses  worth  $310,  paying  the  balance  in 
money.     How  much  money  did  he  pay  ?  $57,625. 

41.  A  farmer  carried  some  pork  to  market,  which  he  sold  for 
$57.62|-,  and  some  poultrj'-,  which  brought  him  $4.18f.  He  paid 
out  $13.50  for  a  coat,  $4.48  for  some  groceries,  and  $29.74  for  a  bill 
of  hardware.     How  much  money  had  he  left  ?  $14.09^. 

42.  A  young  man  bought  a  farm  of  84  acres,  at  $75  an  acre,  and 
made  a  cash  payment  on  it  of  $1,750.  How  much  did  he  run  in 
debt  ?  $%550. 

43.  A  man  sold  a  quarter  of  beef,  which  weighed  156  pounds,  at 
$.08^-  per  pound,  and  expended  the  money  for  nails,  at  $.05^-  per 
pound.     How  many  nails  did  he  purchase  ?  2S4  pounds. 

44.  The  salary  of  the  President  of  the  United  States  is  $25,000. 
How  much  is  that  per  day,  allowing  365  days  to  the  year,  and 
deducting  the  Sundays  ? 

45.  I  bought  a  lot  of  teas  for  $376.75,  and  paid  $31.18f  for 
transportation  on  them.  For  how  much  must  I  sell  them  to  make 
$103.12^  ? 

46.  A  druggist  bought  7  barrels  of  turpentine,  each  containing 
31.5  gallons,  at  $1.37|-  per  gallon.    Wliat  did  the  whole  cost  ? 


102 


DECIMALS, 


What  is  the  sum  of  each  of  the  followiaj 
(47) 


BOOT      AND     SIIOK     TKADE. 
ONE  day's  bales. 


1  Pair  Calf  Boots,  .     . 

.$6.50 

1     '' 

Stoga,     .     .     . 

.  4.75 

1     " 

Coarse,    ,     .     . 

.  2.25 

1     " 

Ladies'  Gaiters, 

.  2.75 

1     " 

Misses'       "     . 

.  1.75 

1     " 

"        Slippers, 

.  1.25 

1     " 

Children's  Shoes, 

.     .5625 

1     " 

Gents'  Slippers, 

.  2.25 

1     " 

Ladies'  Rubbers, 

.     .875 

1     ^' 

Boys'  Boots,     . 

.  2.1875 

1     " 

Misses'  Rubbers, 

.     .75 

Renai 

rinsf" 

.  3.83 

abstracts  of  business ' 
(48) 


DRY      GOODS      TKADK. 
ONE   week's   sales. 


Monday,  .  . 
Tuesday, .  . 
Wednesday, 
Thursda}'',  . 
Friday, .  .  . 
Saturday,.  . 


Cash,  . 

Credit, 

Cash,  . 

Credit, 

Cash,  . 

Credit, 

\  Gash,  . 

(  Credit, 

(  Cash,  . 

1  Credit, 

{  Cash,  . 

(  Credit, 


.$39.24 
.  23.19 
.  61.73 
.  12.48 
.  71.04 
.  56.31 
.  58.98 
.  00 
.  49.06 
.  87.50 
.  65.81 
.129.17 


49.  A  house  agent  rents  7  tenements  at  $1.12j  a  week,  5  at  §1.25, 
and  11  at  $1.50.  What  do  the  rents  amount  to  in  a  year,  or  52 
weeks  ?  $1592.50. 

50.  A  merchant  bought  3  barrels  of  sugar,  containing,  respec- 
tively, 236,  249,  and  261  pounds,  at  $.09|^  per  pound.  What  was 
the  amount  of  the  bill  ?  $70.87. 

51.  If  5  tons  of  coal  are  equal  to  9  cords  of  wood  for  fuel,  and  a 
family  bums  31.5  cords  of  wood  in  a  year,  how  much  will  they  save 
by  changing  from  wood  to  coal,  when  wood  is  worth  $4.25  per 
cord,  and  coal  $6.80  per  ton  ?  $14.87^-  a  year. 

52.  The  rates  of  telegraphing  from  New  York  to  Washington  are 
50  cents  for  the  first  25  words,  and  5  cents  for  each  additional 
word.  At  these  rates,  what  will  be  the  cost  of  sending  a  telegram 
of  117  words  ?  $5.10. 

53.  A  man  bought  3  80-acre  lots,  and  1  40-acre  lot,  of  Govern- 
ment land,  at  $1.25  an  acre.  He  sold  one  half  the  land  at  3  times, 
and  the  balance  at  4  times,  its  original  cost.  For  how  much  did 
he  sell  the  land  ?  $1,225. 

54.  A  commission-merchant  in  Dubuque  shipped  17  tons  of 
prairie  fowls  to  Philadelphia,  where  they  were  sold  at  $.145  per 
pound.  How  much  did  the  shipment  amount  to,  a  ton  being  2,000 
pounds?  $4,930. 


MEASUREMENTS. 


103 


SECTION  VIII. 
sir^BACBS  ^JV2>  sozi^s, 

IJEiniN  ITIONS. 
165.  A  Zfine  is  length  or  distance. 
168.  A  Siraig?it  Jjine  is  the  shortest 
distance  between  two  points.  A- 

167.  A  'Pe7''pe7idicu2ar  is  a  line  which 

stands  upon  another  without  inclining  to  either  side.   Thus, 
the  hne  GB  is  perpendicular  to  the  hne  AB. 

168.  An  Angle  is  the  difference  of  direc- 
tion of  two  lines  that  meet  in  a  point ;  as,  the 
opening  between  the  lines  AB  and  BG. 

160.  A  ^zg/a  Angle  is  one      ^ 
formed  by  two  lines  perpendic- 
ular to  each  other.     Thus,  the 
angles  ABG  and  DEF  are  right 
angles. 

170.  A  Surface y  or  StiperJlcieSy  is  a  figure  that  has 
length  and  breadth. 

171.  A  liecta7igle  is  a  four- 
sided  figure  having  only  right  an- 
gles. Thus,  the  surfaces  ABGD 
and  EFGH  are  rectangles. 

172.  A  Square  is  a  figure  bound- 
ed by  four  equal  sides,  and  having  four  right  angles. 

173.  A  Square  IncJi  is  a  square 
1  inch  long  and  1  inch  wide. 

174.  A  Square  F'oot  is  a  square 
1  foot  long  and  1  foot  wide  ;  a  Square 
^od  is  a  square  1  rod  long  and  1 
rod  wide  ;  and  a  Squa?^e  J)file  is 
a  square  1  mile  long  and  1  mile  wide. 


B     E 


1  Square  Inch^ 


1  Inch  long 


104 


DECIMALS. 


t 

y      ji 

w 

175.  Area  is  tlie  extent  of  any  limited  surface.  Thus, 
if  a  figure  extends  over  a  surface  of  15  square  inches,  its 
area  is  15  square  inches. 

176.  A    Solid    or  E      D  Q  P 
^ody  is  a  figure  that 
has  length,  breadth,  and 
thickness. 

177.  A  :Eecia7igti- 
tar  Solid  is  a  body 
that   has    six    sides   or 

surfaces,  each  of  which  is   a  rectangle.     Thus,  the  solids 
AJWDEF  and  MNOPQR  are  rectangular  solids. 

178.  A  Cube  is  a  body 
bounded  by  six  equal  square 
sides  or  surfaces. 

179.  A  Czibtc  Inch  is 
any  body  or  portion  of 
space  1  inch  long,  1  inch 
v/ide,  and  1  inch  thick. 

180.  A  Cubic  I^oot  has 

six  equal  surfaces  each  1 
foot  square  ;  and  a  Cubic 
Yard  has  six  equal  sur- 
faces each  1  yard  square. 

Notes.— 1.  Length,  width,  and  thickness  are  called  Dimendom. 

2.  A  line  has  one  dimension,  length;  a  surface  has  two  dimensions, 
length  and  width ;  and  a  body  has  three  dimensions,  length,  width,  and 
thickness. 

18L  Capacity  is  the  extent  of  any  body  or  any  portion 
of  space  having  length,  v^idth,  and  thickness.  Thus,  if  a 
body  or  a  portion  of  space  occupies  15  cubic  feet,  its  capac- 
ity is  15  cubic  feet. 

Note. — Areas  and  capacities  are  also  called  Contents. 

182.  j^xte7ision  is  either  length,  area,  or  capacity. 


MEASUREMENTS 


105' 


Measzeres  of  Extension   are  of   the    three   kinds 
named  in  the  three  following  definitions  ; 

183.  Z/inear  JKeasure  is  the  measure  of  lines. 

184.  Superficial  J^feasure^  or  Square  J)feasu7^ey 

is  the  measure  of  surface. 

185.  Solid  Measure y  or  Cubic  Measure ^  is  the 

measure  of  capacity. 

C-A.SEJ    I. 
Measurement  of  Surface. 


186.  Ex.  How  many  square 
inches  in  one  side  of  a  piece  of 
paper  7  inches  long  and  5  inches 
wide? 

Explanation. — If  on  the  paper 
you  draw  lines  just  1  inch  apart, 
both  lengthwise  and  crosswise,  the 
surface  of  the  paper  will  be  divided  into  squares  each  1 
inch  long  and  1  inch  wide.  Since  in  each  of  the  5  rows 
there  are  7  square  inches,  there  are  in  all  5  times  7  square 
inches,  or  35  square  inches.  We  see  from  the  figure  that 
there  are  5  rows  of  7  square  inches  each,  or  7  rows  of  5 
square  inches  each.     (See  80,  V.)     Hence, 

The  number  of  units  in  the  area  of  any  right-angled  surface 
is  equal  to  the  number  of  units  in  the  product  of  its  two  dimeiz- 
sions. 

187.  Ex.  The  contents  of  a  certain  field  are  1,152  square 
rods,  and  the  field  is  36  rods  long.  How  many  rods  wide 
is  it? 

Explanation. — Since  the  field  is  3G  rods 
long,  there  are  38  square  rods  in  a  strip  1  rod 
wide.  And  since  there  are  1,152  square  rods 
in  the  field,  there  must  be  as  many  strips, 
each  1  rod  wide,  as  the  number  of  times  3(1 


SOLUTION. 

1153 [30 

72 

72 


106  DECIMALS. 

square  rods  are  contained  in  1,152  square  rods,  which  is 
32.  Since  each  strip  is  1  rod  wide,  the  lot  must  be  32  rods 
"v\dde.     Hence, 

Either  dimension  of  a  right-angled  surface  is  equal  to  the 
quotient  obtained  by  dimding  the  area  by  the  other  dimension. 

188.  Upon  the  principles  deduced  in  Arts.  186,  187,  is 
based  the 

^ute  for  Jifeasitremeni  of  ^ectajigiilaQ"  Surfaces. 

L  To  find  the  area. 

Multiply  the  length  by  the  breadth, 

II.  To  find  either  dimension. 

Divide  the  area  by  the  other  dimension, 

Pit  Oli  IjEMS. 

1.  If  upon  a  blackboard  19  feet  long  and  5  feet  wide,  I  draw 
lines  1  foot  apart,  both  lengthwise  and  crosswise  of  the  board,  into 
how  many  strips,  lengthwise,  wi.ll  I  divide  the  surface  of  the  board  ? 
How  many  square  feet  will  there  be  in  1  strip  ?  How  many  square 
feet  on  the  surface  of  the  board  ?  95. 

2.  How  many  square  rods  in  a  garden  8  rods  long  and  6  rods 
wide? 

3.  My  slate  is  12  inches  long  and  8.5  inches  wide.  How  many 
square  inches  on  one  side  ?  102. 

4.  How  many  yards  of  carpeting  will  it  take  to  carpet  a  room 
11.5  yards  long  and  5.5  yards  wide  ?  63.25. 

Note.— Numbers  expressing  width  and  length  are  frequently  written 
with  the  word  "  by,"  or  the  sign  of  Multiplication  between  them.  Thus, 
7  by  9  inches,  or  7  X  9  inches,  means  7  inches  wide  and  9  inches  long. 

5.  Hov/  many  square  inches  in  a  pane  of  9  x  14  window-glass  ? 

6.  My  hall  is  2.75  by  8.5  yards.  How  much  must  I  pay  for  oil- 
cloth to  cover  the  floor,  at  $1.12^  per  yard  ? 

7.  How  many  square  feet  in  a  city  lot  75  x  125.3  feet  ? 

8.  How  many  square  rods  in  a  plat  of  ground  5.5  rods  square  ? 

9.  The  area  of  a  door-way  31  inches  wide  is  2,604  square  inches. 
Wliat  is  its  height  ?  SJi,  inches. 


MEASUREMENTS. 


107 


10.  A  farm  in  the  form  of  a  rectangle  is  80  rods  wide,  and  con- 
tains 19,200  square  rods.     What  is  its  length  ?  ^J^O  rods. 

11.  The  ceiling  of  a  room  is  18  feet  long,  and  its  contents  are  288 
square  feet.     How  wide  is  it  ?  16  feet. 

12.  A  carpenter  put  3,375  feet  of  inch  boards  into  the  floor  of  a 
church  45  feet  wide.     What  was  the  length  of  the  church  ?   75  feet. 

13.  I  have  16,875  strawberry  plants,  in  75  equal  rows.  Hovr 
many  plants  in  each  row  ?  225. 

C^SE    II. 
Measurement  of  Capacity. 
189.  Ex.  How  many  cubic  feet  in  a  block  of  stone  5  feet 
long,  4  feet  wide,  and  3  feet  thick  ? 

Explanation. — If  5  blocks, 
eacli  containing  1  cubic  foot, 
be  placed  side  by  side,  they 
will  form  a  row  5  feet  long, 
1  foot  wide,  and  1  foot  thick.  ] 
If  4  such  rows  be  placed 
side  by  side,  they  will  form 
a  layer  5  feet  long,  4  feet 
wide,  and  1  foot  thick. 

If  3  such  layers  be  placed  one  exactly  upon  the  other,  they 
will  form  a  pile  5  feet  long,  4  feet  wide,  and  3  feet  thick. 
Hence, 

4  times  5  cubic 
feet  =  20  cubic 
feet,  the  number 
in  1  layer  ;  and 
3  times  20  cubic 
feet  =  60  cubic 
feet,  the  number 
in  the  pile  or  in  the  block. 

There  are  as  many  cubic 
feet  in  one  row  of  these 
blocks   as   the   pile  is  feet 


SOLtlTIOJf. 

5  cubic  feet. 
J. 
2  0  cubic  feet. 


6  0  cubic  feet. 


108 


D  E  C  I  M  AL  S. 


long,  as  many  rows  of  blocks 
in  one  layer  as  the  pile  is 
feet  wide,  and  as  many  lay- 
ers in  the  pile  as  the  j)ile  is 
feet  high. 

The  number  of  units  in  the 
capacity  of  any  right-angled 
body  or  portion  of  ^ace  is 
equal  to  the  ^lumber  of  units  in 
the  product  of  its  three  dimen- 
sions. 

190.  Ex.  A  paper-box  maker  made  a  package  of  432 
boxes,  putting  8  boxes  in  each  row,  and  6  rows  in  each 
layer  or  tier.     How  many  boxes  high  was  the  package  ? 

Explanation. — Since  in  1  row 
there  were  8  boxes,  in  the  6  rows 
of  1  layer  or  tier  there  were  6      6  x  8  =  JfS 
times  8  boxes,  or  48  boxes.   Since 
in  the  whole  package  there  were 

432  boxes,  and  in  1  layer  48  boxes,  there  were  in  the  pile  as 
many  layers  as  the  number  of  times  48  is  contained  in  432, 
which  is  9.  As  there  were  9  layers,  each  one  box  high,  the 
package  was  9  boxes  high.     Hence, 

Any  one  of  the  three  dimensions  of  a  right-angled  body  or 
portion  of  space  is  equal  to  the  quotient  obtained  by  dividing  the 
capacity  by  the  product  of  the  other  two  dimensions. 

191.  Upon  the  principles  in  Arts.  189,  190,  is  based  the 
^ule  for  Jifeasurement  of  (Rectangular  Solids » 

I.  To  find  the  capacity. 
Multiply  the  length,  width,  and  thickness  together. 

n.  To  find  any  one  of  the  three  dimensions. 
Divide  the  capacity  by  the  product  of  the  other  two  dimen- 
sions. 


SOLUTION. 

J^32 
Jf.32 


9 


ME  ASUEEMENTS.  109 


14.  A  pile  of  bricks  consists  of  7  layers,  and  each  layer  contains 
8  rows  of  9  bricks  each.     How  many  bricks  in  the  pile  ?         50^. 

15.  How  many  blocks,  each  measuring  1  cubic  inch,  can  you  put 
into  a  box  7x6x4  inches  inside  ? 

IG.  An  embankment  12  feet  high  and  4  feet  thick  contains  6,000 
cubic  feet.     How  long  is  it  ?  125  feet. 

17.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high,  con- 
tains 1  cord.     How  many  cubic  feet  in  a  cord  ? 

18.  How  many  cubic  feet  in  a  stick  of  timber  85  feet  long,  3  feet 
wide,  and  1.5  feet  thick  ?  105. 

19.  The  contents  of  a  pile  of  vv^ood  4  feet  wide  and  5  feet  high 
are  1,280  cubic  feet.     "What  is  its  length  ?  GJi.feet. 

20.  How  many  cubic  yards  of  earth  will  be  removed  in  digging 
a  cellar  27  feet  long,  24  feet  wide,  and  7  feet  deep,  there  being  27 
cubic  feet  in  a  cubic  yard  ?  1G8. 

21.  What  is  the  capacity  of  a  space  22.5  feet  long,  6.4  feet  wide, 
and  3.25  feet  deejj  ?  J!i,68  cvMcfeet. 

22.  A  music  dealer  found  that  a  packing  box  that  would  hold  a 
melodeon  which  was  18.5  inches  wide  and  28  25  inches  high, 
must  have  a  capacity  of  21323.1  cubic  inches.  Allowing  the  lum- 
ber to  be  1  inch  thick,  what  were  the  outside  dimensions  of  the 
box  ?  20.5  ly  30.25  hj  ^2.8  inches. 

23.  In  a  granary  is  a  bin  7  x  8  x  2.25  feet.  What  is  its 
capacity  ? 

24.  The  walls  of  a  stone  building  45  feet  long  and  24  feet  wide, 
are  36  feet  high  and  1  foot  thick.  How  many  cubic  feet  of  ma- 
sonry in  the  walls,  no  allowance  being  made  for  openings  ?     Jf,82Jf. 

25.  How  many  cubic  inches  in  a  cubic  foot  1 

26.  How  many  cubic  yards  in  a  cubic  mow  of  hay  which  meas- 
ures 1  rod,  or  5.5  yards,  in  each  of  its  three  dimensions  ?  166.375. 

27.  Two  of  the  dimensions  of  a  stone  column  which  contains 
196.8  cubic  feet,  are  12.3  feet  and  4  feet.  What  is  the  shape  of  one 
end  of  the  columu  ?  It  w  square. 


110  DECIMALS. 

SECTION  IX. 

1.  A  contractor  employed  37  laborers  56  days,  13  laborers  84 
days,  13  laborers  43  days,  and  17  laborers  21  days.  What  was  the 
total  amount  of  their  wages,  at  $.87i  per  day  ?  $3532.37 f 

2.  A  grocer  buys  4  barrels  of  kerosene,  each  containing  31.5 
gallons,  for  $55.12^,  and  he  wishes  to  sell  it  at  a  profit  of  $.18|- 
per  gallon.    At  what  price  per  gallon  must  he  sell  it  ? 

3.  At  $7.87|  per  bushel,  how  many  bushels  of  grass  seed  can  be 
bought  for  $66.93f  ?  ^-^. 

4.  A  fish  dealer  has  45  barrels  of  mackerel,  which  he  wishes  to 
repack  into  kits  holding  12.5  and  25  pounds  each,  and  to  use  an 
equal  number  of  each  size.     How  many  kits  must  he  use  ?      480. 

5.  If  67.5  bushels  of  oats  are  required  to  feed  one  horse  through 
the  winter,  how  many  horses  can  be  wintered  on  950  bushels  ? 

14j  with  5  JnisheU  left. 

6.  If  in  1  hour  1,354  gallons  of  water  run  into,  and  1010.8  gal- 
lons run  out  of,  a  reservoir  which  will  hold  23,381  gallons,  and  the 
reservoir  now  contains  12999.2  gallons  of  water,  in  how  many  hours 
will  it  be  full?  30.25. 

7.  The  terms  of  a  weekly  newspaper  are,  to  single  subscribers, 
$1.50 ;  clubs  of  three,  $3.75 ;  clubs  of  five,  $5.00 ;  clubs  of  ten, 
$8.75.  The  paper  has  694  single  subscribers,  63  clubs  of  three,  47 
clubs  of  five ;  34  clubs  of  ten,  and  a  free  exchange  list  of  50  cojjies. 
What  is  the  total  circulation,  and  what  are  the  receipts  from  sub- 
scriptions? Circulation,  1,508 ;  receipts,  $1809.75. 

8.  A  ton  of  iron  ore  from  Iron  Mountain  yields  .50  of  a  ton  of 
pure  iron.     How  much  iron  will  736.72  tons  of  ore  yield  ? 

9.  A  drover  bought  69  beeves  at  $28.75  a  head,  and  sold  42  of 
them  at  $36.5  a  head,  and  the  rest  at  $37.75  a  head.  How  much 
did  he  gain  by  the  transaction  ?  $568.5. 

10.  A  and  B  start  from  the  same  place  at  the  same  time,  and 
travel  in  opposite  directions,  A  traveling  at  the  rate  of  23.16  miles 
per  day,  and  B  at  the  rate  of  19.21  miles  per  day.  How  far  wall 
they  be  apart  at  the  end  of  17.4  days  ?  737.238  miles. 


REVIEW    PROBLEMS.  Ill 

11.  In  a  scliool-room  are  8  rows  of  double  desks,  and  7  desks  in  a 
row.  How  many  pupils  can  be  seated  at  the  desks  in  the  school- 
room ?  112. 

13.  A  copper-plate  engraver  bought  a  plate  of  copper  16.35  by 
25.3  inches,  @  $.03  jDer  square  inch.    How  much  did  it  cost  him  ? 

13.  I  bought  a  boat  load  of  wood  for  $183,  and  by  retailing  it  at 
$4.50  a  cord,  I  gained  $70.  How  many  cords  of  wood  were  there 
in  the  load  ?  56. 

14.  I  borrowed  some  money,  and  paid  $41.79  for  the  use  of  it, 
paying  $.105  for  the  use  of  each  dollar  borrowed.  How  much 
money  did  I  borrow  ?  $398. 

15.  A  government  township  of  land  is  6  miles  square.  How 
many  square  miles  does  it  contain  ? 

IG.  My  garden,  which  is  square,  is  inclosed  by  a  tight  board 
fence  8.5  feet  high,  and  the  fence  contains  2,805  feet  of  lumber. 
What  is  the  length  of  one  side  of  my  garden  ?  82.5  feet. 

17.  One  year  a  farmer's  account  of  grain  sold  was  as  follows  : 

SJ'/f^u^/fe^  e>a/j @  /  .  (f^S 

m    '^    'tfui^j^ ''  /^/c^ 


18.  How  many  square  feet  in  the  four  walls  of  a  room  18  feet 
long,  13.25  feet  wide,  and  9.5  feet  high  ? 

19.  A  publisher's  expenses  in  publishing  an  edition  of  3,000 
copies  of  a  certain  book  are,  for  stereotyping,  $515  ;  paper,  $365  ; 
binding,  $370  ;  engraving,  $80.  What  must  be  the  retail  price  of 
the  book,  that  the  author  may  receive  a  copyright  of  8  cents  per 
copy,  the  publisher's  profit  be  30.5  cents,  and  the  retail  bookseller's 
profit  30  cents  per  copy  ?  $1.25. 

30.  Find  the  amount  of  six  thousand  one  hundred  nineteen  mill- 
ionths,  four  hundred  eight  and  twenty-six  thousandths,  two  million 
twenty  thousand  two  hundred  and  seven  hundred  three  ten-thou- 
sandths, and  thirty  thousand  sixty-five  hundred-millionths. 

Two  million  twenty  thousand  six  hundred  eight  and  ten  million  two 
hundred  seventy-one  thousand  nine  hundred  sixty-Jive  hundred-millionths. 


112  DECIMALS. 

21.  A  steamship  made  a  voyage  in  eiglit  days,  sailing  the  follow- 
ing distances :  217  miles,  198.85  miles,  246.7856  miles,  208  miles, 
227.6987  miles,  200.045  miles,  241  miles,  and  205.08675  miles. 
What  was  the  length  of  the  voyage  ?  174446G05  miles. 

22.  A  farmer  bought  a  yoke  of  oxen,  which  he  fattened,  slaugh- 
tered, and  sold.  The  following  is  taken  from  his  account  of  the 
entire  transaction : 

Disbursemerds. 

^a(c//oi    /^o/e  oa;en '^/y/6>.  00 

Sodi( o/ /a'^/encna  >//{&  ,jame //.SO- - ^/S% SO 


Receipts. 

-^  ^{'nc/auui/eiJ/  /S^S,   /2^  //S",  anc/ 7^6'0  Aounc^^  @,^. /f 
//Aie         ''         /S//,  /i'^  /SJ,    -^    /SP      '^         ^'  ./Si- 

S  '/(<c/e^,    S/!//iOunc/d ''   . /^ 

'^^  /iounc/j  /a/tfozo- ^^  .  /Sj; 

^^//uoA/d ^d'S.SS 


23.  From  a  stock  of  214  tons  of  coal,  a  dealer  sold  in  one  week 
6.65  tons,  9  tons,  8.775  tons,  9.27  tons,  5.45  tons,  and  7.125  tons. 
How  much  coal  had  he  left  in  his  yard  ?  167.73  tons. 

24.  Two  men  start  from  the  same  place,  at  the  same  time,  and 
travel  in  the  same  direction,  one  at  the  rate  of  23.75  miles,  and  the 
other  of  19.5  miles  per  day.  In  how  long  a  time  will  they  be  64.6 
miles  apart  ? 

25.  How  far  apart  will  they  be  in  the  same  time,  if  they  travel  in 
opposite  directions  ?  657.^  miles. 

26.  A  merchant  and  a  farmer  bartered.  The  farmer  sold  to  the 
merchant  37.25  pounds  of  butter  @  $.27,  and  21.5  dozen  of  eggs 
@  $.19 ;  and  the  merchant  sold  to  the  farmer  12  yards  of  shirting 
®  $.25,  and  13.5  yards  of  calico  ®  $.28.  The  balance  was  paid  in 
money.    How  much  was  paid^  and  who  paid  it  ? 

27.  How  many  reams  of  commercial  note  paper  each  3.5  x  5  x  8 
inches  in  size,  can  be  packed  in  a  box  the  inside  dimensions  of  which 
are  14  x  20  x  32  inches  ?  64. 


SECTIOH  I. 

^BFIJVI  TIOjYS. 

193.  Some  articles  are  bought  and  sold  by  the  quart  or 
gallon ;  some  by  the  peck  or  bushel ;  some  by  the  foot  or 
yard ;  some  by  the  acre  ;  some  by  the  cord ;  some  by  the 
pound  or  ton,  and  so  on. 

194.  Measure  is  that  by  which  extent,  dimension,  or 
quantity  of  matter  is  ascertained,  whether  it  be  length, 
breadth,  thickness,  or  amount. 

195.  }feig?it  is  a  measure  of  the  amount  of  matter,  or 
the  quantity  of  heaviness,  in  a  body. 

196.  Weight  and  measure  are  determined  by  processes 
called  Weighing  and  Measuring,  which  consist  in  compar- 
ing the  thing  to  be  weighed  or  measured  with  some  fixed 

standard.      See  Manual, 

197.  ^enominaiioTi  is  the  name  of  the  unit  of  a  con- 
crete number  ;  as,  gallon,  foot,  pound,  hour,  dollar. 

198.  A  ^e?iominaie  jVicmber  is  a  number  apphed  to 
a  denomination  ;  as,  9  quarts,  4  feet,  $7. 

199.  A  Simple  JVuJuder  is  an  abstract  number ;  as, 
G,  43,  915  ;  or  a  concrete  number  of  but  one  denomination  ; 
as,  78  men,  324  miles. 

200.  A  Compoimd  JYumber  is  a  number  expressed  in 
two  or  more  denominations ;  as,  4  pounds  10  ounces,  15 
gallons  3  quarts  1  pint. 

201.  JligJier  ^e7i07mnaHo7is  are  those  which  ex- 
press the  greater  amount  or  quantity. 


114  COMPOUND    NUMBERS. 

202.  Z/07Per  ^enomi?iaHo7is  are  those  whicli  express 
the  less  amount  or  quantity.  Thus,  a  peck  is  a  higher 
denomination  than  a  quart,  and  a  lower  denomination  than 
a  bushel. 

Notes. — 1.  A  denominate  number  may  be  an  integer,  as  7  busliels ;  a  deci- 
mal, as  .75  of  a  mile ;  a  mixed  number,  as  5.135  gallons ;  or  a  compound 
number,  as  4  days  9  hours  20  minutes. 

2.  Compound  numbers  are  sometimes  called  denominate  numbers ;  but 
tbc  term  denominate  number  properly  belongs  to  a  concrete  number  of  one 
denomination. 

203.  A  Table,  in  Compoimd  Numbers,  is  a  regular 
arrangement  of  the  denominations  used  to  express  a  com- 
pound number. 

204.  A  ^Denominate  ^7^^'/ is  one  of  any  denomination ; 
as  1  pound,  1  foot,  1  quart. 

205.  A  Sta?idard  Unit  of  a  table  is  that  unit  which  law 
or  custom  has  established  as  the  one  from  which  the  other 
denominations  in  the  table  are  determined.  Thus,  the 
standard  unit  of  length  is  the  yard;  the  foot  and  inch  being 
obtained  by  dividing  this  standard  unit,  and  the  rod  and 
mile  by  multiplying  it. 

206.  In  computations  in  compound  numbers,  it  is  often 
necessary  to  change  units  of  higher  denominations  to  those 
of  lower,  as  gallons  to  quarts  or  pints  ;  or  units  of  lower 
denominations  to  those  of  higher,  as  feet  to  rods  or  miles. 

deductions  are  processes  of  changing  numbers  from 
one  denomination  to  another  without  changing  values. 
They  are  of  two  kinds,  Bediiciion  Descending  and  Reduction 
Ascending. 

207.  deduction  descending  is  the  process  of  chang- 
ing numbers  from  higher  to  lower  denominations  ;  and 

208.  ^educHo7i  Ascending  is  the  process  of  chang- 
ing numbers  from  lower  to  higher  denominations. 

209.  The  scale  of  decimal  numbers  being  decimal,  the 
successive  orders  of  units  increase  and  decrease  uniformly 


NOTATION    AND    KEDUCTIONS. 


115 


by  10  ;  wliile  the  scales  of  most  compound  numbers,  being 
varying,  the  successive  orders  of  units  have  no  regular  rate 
of  increase  and  decrease.  From  this  difference  in  their  scales 
arises  the  only  difference  in  computations  in  the  two  classes 
of  numbers. 

210.  The  different  orders  of  units  of  a  compound  number, 
like  those  of  an  integer,  increase  in  value  from  right  to  left, 
the  higher  denominations  being  written  at  the  left. 

211,  the  denominations  of  compound  numbers  are  gen- 
erally abbreviated,  as  shown  in  the  tables,     see  Manual. 


SECTION  II. 

JVOTA.TIOJV  ;^JVD  "EJSDUCTIOJVS. 

212.  Money  is  the 

legal  or  recognized  stand- 
ard of  value.  It  consists 
of  coins,  made  of  gold, 
silver,  or  other  metal. 

213.  Treasury  JVoles 

are  notes  or  bills  issued 
by  the  General  Govern- 
ment ;  and 

214.  ^an/c  JVotes  ov 
^an^  :Bills  are  notes 
or  bills  issued  by  a  bank- 
ing company. 

215.  ^aper  Money  consists  of  notes  or  bills  issued  by 

the  Government  or  a  bank. 

Notes.— 1.  Treasury  notes  are  payable  to  the  bearer,  at  the  General  Treas- 
ury of  the  United  States,  at  a  time  specified  in  them. 

2.  Bank  notes  are  payable  to  the  bearer,  at  the  bank,  on  demand. 

3.  Paper  money  is  a  substitute  for  coin,  and  is  regarded  and  circulated 
as  money. 


116  COMPOUND    NUMBERS. 

216.  Ctirre7icy  is  the  coin,  treasury  notes,  bank  notes, 
and  other  substitutes  for  money,  or  recognized  representa- 
tives of  value,  in  circulation  in  trade  and  commerce. 

Notes.— 1.  Coin  is  commonly  called  Specie  Currency^  or  Specie;  and  treas- 
ury and  bank  notes  are  called  Paper  Currency. 
2.  Every  civilized  nation  has  its  own  kind  of  money. 

Table  I,— United  States  Money, 

217.  I7?iited  States  Mo7iey^  or  Federal  Money,  is 
the  money  of  the  United  States.  Its  denominations,  as 
estabhshed  by  the  General  Government,  are  eagles,  dollars, 
dimes,  cents,  and  mills. 


10  mills    are  1  cent. 
10  cents     "    1  dime. 
10  dimes    "    1  dollar. 
10  dollars  "    1  eaofle. 


1  eagle    is  10  dollars. 
1  dollar  "10  dimes. 
1  dime    "10  cents. 
1  cent      "  10  mills. 


Scale. — Decimal,  or  uniformly  10. 

Note.— The  subject  of  United  States  money  has  already  been  fully  treated 
in  Chap.  II.,  Sec.  VII.  The  Government  table  is  given  here,  for  the  purpose 
of  presenting  all  the  tables  of  Compound  Numbers  in  the  same  Chapter. 

Table  II, — Canada  Money, 

218.  Canada  J)fo?zey  is  the  money  of  Canada,  gr  The 
New  Dominion.     Its  denominations  are  dollars  and  cents. 

100  cents  are  1  dollar.      ]      1  dollar  is  100  cents. 
Scale. — Uniformly  100. 
The  coins  are  the  5-cent  piece,  10-cent  piece,  and  20-cent  piece. 

Note.— In  business  transactions  20  cts.  (cents)  are  called  1  s.  (shilling), 
and  5  shillings  1  dollar. 

Table  III,— English  3Ioney, 

219.  SJnglish  Mo7iey,  or  Sterling  Money,  is  the 
money  of  Great  Britain.  Its  denominations  are  pounds, 
shillings,  pence,  and  farthings. 

4  far.,  or  qr.  (farthings)  are  1  d.  (penny.)  £1  is  20  s. 
13  d.  (pence)  "    1  s.  (shilling.)      1  s.    "  12  d. 

20  s.  "  £1     (pound)         Id."    4  far.,  or  qr. 

Scale.— Ascending,  4,  12,  20 ;  descending,  20,  12,  4. 


NOTATION    AND    REDUCTIONS. 


117 


Notes. — 1.  The  abbreviation  £  is  always  written  before,  and  the  other 
abbreviations  are  written  after  the  numbers  to  which  tliey  give  denomina- 
tion or  name.    Thus,  3  pounds  5  shillings  4  pence  is  written  £3  5  s.  4  d. 
2.  The  coins  are  of  gold,  silver,  and  copper. 
I.  Gold  Coins.  —  Sovereign  =  £1,  half-sovereign  =  10  s.,  guinea  =  21s.  = 

£1 1  s.,  and  half-guinea  =  10  s.  6  d. 
II.  Silver  Coins, — Crown  =  5  s,,  half-crown  =  2  s.  6d.,  shilling  =  12  d.,  and 
sixpence  =  6  d. 
III.  Copper  Coins.— Fenny  =  4  far.,  half-penny  =  2  far.,  and  farthing. 

JEXEJRCISJES. 

1.  Read  £4  7  s.  6  d. ;  £17  4  s.  3  d.  2  far. ;  18  s.  9  d.  1  qr. 

2.  Write  25  pounds  10  shillings ;  £  eighteen  three  s. 

3.  Write  217  pounds  9  shillings  1  penny  2  farthings. 

4.  Write  £  eight  seventeen  s.  six  d.  three  far. 

5.  Write  17  shillings  3  farthings ;  11  pounds  9  pence  1  farthing. 

220.  Measures  of  Ca- 
pacity are  of  three  kinds, 
Liquid  Measure,  Dry 
Measure,  and  Cubic 
Measure. 

Note. — Cubic  Measure  is  also 
classed  under  Measures  of  Ex- 
tension.   (See  228.) 

Table  IV, 
Liquid  Measure. 

221.  The  table  of  liquid 
measure  consists  of  the 
denominations  gallons, 
quarts,  pints,  and  gills. 
These  denominations  are 

used  in  measuring  oil,  molasses,  syrups,  wines,  milk,  and 
other  liquids. 


4  gi.  (gills)  are  1  pt.  (pint.) 
2  pt.  "    1  qt.  (quart.) 

4  qt.  "1  gal.  (gallon.) 

Scale. — Ascending  and  descending,  4,  2,  4. 


1  gal.  is  4  qt. 
1  qt.  "  2  pt. 
1  pt.    "  4  gi. 


118  COMPOUND    NUMBERS. 

In  estimating  the  capacity  of  cisterns,  reservoirs,  etc., 

31.5  gal.  are  1  bar.,  or  bbl.  (barrel.) 
63  gal.       "    1  hbd.  (hogshead.) 

Notes.— 1.  The  barrels  and  hogsheads  used  for  commercial  purposes  are 
not  fixed  measures ;  the  former  containing  from  30  to  45  gallons,  and  the 
latter  from  60  to  125  gallons. 

2.  Physicians  in  prescribing,  and  apothecaries  in  mixing  medicines  that 
are  liquids,  divide  the  gallon  according  to  the  following 

APOTHECAEIES'    FLUID  MEASUEES. 

60  minims  (or  drops)  are  1  fluid  drachm. 

8  fluid  drachms  "    1  fluid  ounce. 

16  fluid  ounces  "   1  pint. 

8  pints  "   1  gallon. 

3.  Thsy  also  use  the  following  measures,  from  vessels  in  common  use : 
4  tea-spoons  are  1  table-spoon,  2  table-spoons  are  1  ounce,  2  ounces  are  1 
wine-glass,  2  wine-glasses  are  1  tea-cup,  4  tea-cups  are  1  pint. 

Since  the  vessels  named  are  not  made  of  uniform  size,  the  values  given 
must  necessarily  vary. 

Table  V,—Dry  3Ieasure* 

222.  The  table  of  dry  measure  consists  of  the  denomi- 
nations bushels,  pecks,  quarts,  and  pints.  These  denomi- 
nations are  used  in  measuring  grain,  seeds,  fruits,  berries, 
several  kinds  of  vegetables,  lime,  charcoal,  and  some  other 
articles. 


2  pt.  are  1  qt. 

8  qt.    '•    1  pk.  (peck.) 

4  pk.  "    1  bu.  (bushel.) 


1  bu.  is  4  pk. 
1  pk.  "  8  qt. 
1  qt.  "  3  pt. 


C5CALE— Ascending,  2,  8,  4  ;  descendmg,  4,  8,  2. 

Notes.— 1.  Where  fruit  and  vegetables  are  marketed  by  the  basket  or  bar- 
rel, a  peach  basket  should  hold  16  qt.  or  2  pk.  ;  a  potato  basket,  24  qt.  or 

3  pk. ;  and  a  barrel,  3  potato  baskets.    Barrels  made  for  marketing  pur- 
poses commonly  hold  100  qt. 

2.  In  measuring  grain,  seeds,  peas,  beans,  and  small  fruits,  the  measure 
must  be  even  full.  But  in  measuring  large  fniits,  corn  in  the  ear,  coarse 
vegetables,  and  other  bulky  articles,  the  measure  must  be  heaping  fiiU. 

4  heaped  measures  must  equal  5  even  measures.     See  ManuaL 


NOTATION    AND    REDUCTIONS. 


119 


EXEIt  CIS  ES. 

6.  Read  4  gal.  3  qt.  1  pt.  2  gi. ;  15  gal.  1  pt. 

7.  Read  11  bu.  3  pk.  5  qt.  1  pt. ;  8  bu.  7  qt. 

8.  Write  17  gallons  1  quart  1  pint  3  gills. 

9.  Write  260  bushels  3  pecks  4  quarts  1.5  pints. 

10.  Write  four  gal.  three  qt.  two  gi. ;  3  bar.  15.5  gal. 

11.  Write  eleven  bu.  one  pk.  six  and  seven-tenths  qt. 

12.  Read  13.125  bu. ;  41.75  gal. ;  1  pk.  4.375  qt. ;  1  gal.  1.8  pt. 

223.  Ex.  1.  How  many  pints  are 
17  gallons  ? 

Explanation. — Since  17  gal.  are  17 
times  1  gal.,  and  1  gal.  is  4  qt.,  17 
gal.  are  17  times  4  qt.,  or  68  qt.  ;  and 
since  68  qt.  are  68  times  1  qt.,  and 
1  qt.  is  2  pt.,  68  qt.  are  68  times  2 
pt.,  or  136  pt. 


SOLUTION. 

1  7  gal. 
Jt 
68qt. 

2 

136pt. 
Hence,  17  gal.  =  136  pt. 


Ex.  2.  How  many  quarts  are  8  bu.  3  pk.  4  qt.  ? 

Explanation. — Since  8  bu. 
are  8  times  1  bu.,  and  1  bu.  is 
4  pk.,  8  bu.  are  8  times  4  pk.,  or 
32  pk.,  and  32  pk.  +  3  pk.  are 
35  pk.  Since  35pk.  are  35  times 
1  pk.,  and  1  pk.  is  8  qt.,  35  pk. 
are  35  times  8  qt.,  or  280  qt., 
and  280  qt.  +  4  qt.  are  284  qt. 

In  the  Common  Solution 
we  mentally  added  the  3  pecks 
with  the  pecks  of  the  partial 
result,  and  the  4  quarts  with 
the  40  quarts  of  the  final  re- 
sult. This  manner  of  solution 
is  the  one  in  common  use. 


FULL  SOLUTION. 

8  bu.  8  pk.  Jf.  qt. 

S2  +  3  =  35pTc. 
8 

280+4.  =  28^qt. 

Hence,  8  hi.  3  ph  4qt.  =  28^  qt, 

COMMON  SOLUTION. 

8  bu.  3  pk.  Jj.  qt, 

35pk. 
8 

28Ji.qt. 


120  COMPOUND    NUMBERS. 

In  Ex.  1  we  reduced  gallons  to  quarts,  and  quarts  to 
pints — that  is,  a  higher  to  a  lower  denomination — by  multi- 
plication ;  and  in  Ex.  2  we  reduced  bushels  and  pecks  to 
quarts — that  is,  higher  denominations  to  a  lower  denomi- 
nation— ^by  multipHcation.     Hence, 

A  denominate  number  is  reduced  to  lower  denominations  by 
multiplication. 

rjt  OBL  EMS. 

1.  Reduce  £32  to  shillings.     To  pence.  7,680  d. 

2.  How  many  mills  are  there  in  249  dollars  43  cents  ? 

3.  In  $93  4  s.  17  cts.,  Canada  currency,  how  many  cents  ?  9,397. 

4.  £45  8  s.  6  d.  are  how  many  farthings  ?  JfS,  608. 

5.  At  1  s.  a  yard,  how  many  yards  of  sheeting  can  be  bought  for 
£11  10  s.  ?  230. 

6.  Reduce  9  s.  7.25  d.  to  farthings. 

7.  How  many  pence  in  £231  8  d.  ?  solution  of  tkoblem  t. 

8.  How  many  gills  are  there  in  15  gallons  ?         ^^^lo^^' 

9.  Reduce  27  gal.  3  qt.  2gi.  to  gills.  Jl620s 

10.  A  grocer  has  87  gal.  1.5  qt.  of  cider  12 
which  he  wishes   to   put  into  pint  bottles.        SBJfJ^Sd. 
How  many  bottles  must  he  use  ?             699. 

11.  18  bushels  are  how  many  quarts  ? 

12.  Two  brothers  picked  6.375  bushels  of  berries.  How  many 
quarts  of  berries  had  they  ?  ^OJf. 

13.  A  seedsman  put  up  2  bu.  4  qt.  of  marrowfat  peas  in  pint 
papers.     How  many  papers  did  he  fill  ? 

14.  Reduce  126  bu.  1  pt.  to  pints. 

15.  How  many  quart  boxes  will  be  required  to 
hold  7  bu.  3  pk.  7  qt.  of  strawberries  ?         255. 

16.  £6.5  are  how  many  shillings  ?  How  many 
pence  ? 

17.  A  dealer  in  findings  sold  13  bu.  3  pk.  6  qt. 
of  shoe-pegs  by  the  quart.  How  many  quarts 
did  he  sell?  8065  pt. 


SOLUTION 
PKOBLEM 

\   OP 

14. 

126  hi. 

4 

Ipt. 

50^  pTc. 
8 

4032  qt. 
2 

NOTATION    AND    REDUCTIONS. 


121 


REDTJCTIOIS'   J^SCENIDUNG-, 

224  •  Ex.  1.  How  many  bushels  arc 
192  quarts? 

Explanation. — Since  every  8  qt.  are 
1  pk.,  and  8  qt.  are  contained  in  192 
qt.  24  times,  192  qt.  arc  24  pk.  And 
siQce  every  4  pk.  are  1  bu.,  and  4  pk. 
are  contained  in  24  pk.  6  times,  24 
pk.  are  6  bu. 


19^  qt  [  8  qt. 

2  Jf  times. 

102  qt.  =  24.  pL 

2Jf.pk.  {J{.plc. 

6  times. 

2J^ph.  =  6hu. 

Hence,  192  gt.  =  6  M. 

Ex.  2.  How  many  bushels  are  637  quarts  ? 
Explanation. — Since  every 


8  qt.  are  1  pk.,  and  8  qt.  are 
contained  in  637  qt.  79  times 
with  a  remainder  of  5  qt., 
637  qt.  are  79  pk.  5  qt.  And 
since  every  4  pk.  are  1  bu., 
and  4  pk.  are  contained  in 
79  -pk.  19  times  with  a  re- 
mainder of  3  pk.,  79  pk.  are 
19  bu.  3  pk. 

In  the  Full  Solution  we 
have  written  all  the  numbers 
mentioned  in  the  explana- 
tion ;  but  in  the  Common 
Solution    we   have   written 


FULL  SOLTTTION. 

6  37  qt.  {8  qt. 

7  9  times  and  5  qt.  rem. 
637  qt.  =  7  9p1c.5  qt. 
79pk.  [^pk. 

1  9  times  and  3  ph.  rem. 
7  9pk.  =  19bu.3pk. 
Hence,  637  qt.  =  19  lu.  3  pK  5  qt. 

COMMON   SOLUTION. 

63  7  qt.  { 8  qt. 
7_9ph.5  qt.  {J^pk. 
19hu.3pk. 
Hence,  637  qt.  =  19  hi.  3  p7c.  5  qt. 


only  the  deiiominate  num- 
bers.    This  manner  of  solution  is  the  one  in  common  use. 

In  Ex.  1  we  reduced  quarts  to  pecks,  and  pecks  to  bush- 
els— ^that  is,  a  lower  to  a  higher  denomination — by  division  ; 
and  in  Ex.  2  we  reduced  quarts  to  bushels,  pecks,  and 
quarts — that  is,  a  lower  denomination  to  higher  denomina- 
tions— ^by  division.     Hence, 

A  denominate  number  is  redvxied  to  higher  denominations  hy 
division. 

6 


122  COMPOUND    NUMBERS. 

rjt  obtjEms. 

18.  Reduce  256,327  mills  to  dollars,  cents,  and  mills. 

256  dollars  32  cents  7  mills. 

19.  1680  d.  are  how  many  £  ? 

20.  How  much  will  765  bushels  of  lime  cost  at  1  s.  a  bushel  ? 

21.  How  many  gallons  are  486  pints  ?        60.75^  or  60  gal.  3  qt. 

22.  How  many  times  can  you  till  a  gallon  measure  from  1,024  gills 
of  alcohol  ? 

23.  Reduce  733  pints  dry  measure  to  higher  denominations. 

11  III.  1  pTc.  6  qt.  1  pt. 

24.  How  many  bushels  are  7,380  quarts  of  shelled  com  ? 

227.5,  or  227  lu.  2  pi. 

25.  One  season  a  gardener  sold 

3,975  quart  boxes  of  strawberries.           boljtion  of  pkoblkm  25. 
Jr         ^       ,      1,1     Til        no                3  97  5  qt.\8qt. 
How  many  bushels  did  he  sell  ? ^     ».    :/ 

26.  How  many  gallons  of  sweet- ^  '     "  '  ^"^P  ' 

oil  will  an  apothecary  use  in  filling  ^  ^  -^  ^'^* 

500  gill  bottles  ?  Hence,  3975  qt.  =  12J^  hi.  7  qt. 

15.625,  or  15  gal.  2  qt.  1  pt. 

27.  A  grocer  buys  187  qt.  of  walnuts.     How  many  bushels  does 
he  buy  ? 

28.  A  cask  is  emptied  in  892  minutes,  by  a  pipe  which  discharges 
1  pint  of  water  per  minute.     What  is  the  capacity  of  the  cask  ? 

225.  Upon  the  principles  deduced  in  Arts.  223,  224,  is 
based  the 

^ules  for  ^Reductions  of  Compound  JV^umbers, 

I.  For  Reduction  Descending. 

1.  Multiply  the  number  of  the  highest  denomination  given, 
whether  integer,  decimal,  or  mixed  number,  by  that  number  of 
the  next  lower  denomination  which  equals  1  of  this  higher,  and 
to  the  product  add  the  given  lower  denomination. 

2.  In  the  same  manner,  reduce  this  result  to  the  next  lower 
denomination ;  and  so  continue  until  the  given  number  is  re 
duced  to  the  required  denomination. 


NOTATION    AND    REDUCTIONS.  123 

n.  For  Keduction  Ascending. 

1.  Divide  the  number  of  the  given  denomirmtioriy  whether 
integer,  decimal,  or  mixed  numler,  hy  that  number  of  this  de- 
nomination which  equals  1  of  the  next  higher,  ivriting  the  quo- 
tient as  so  many  of  the  higher  denomination,  and  the  remainder 
as  so  many  of  the  denomination  divided, 

2.  In  the  same  manner,  reduce  this  quotient  to  the  next  higher 
denomination ;  and  so  continue  until  the  given  number  is  re- 
duced to  the  required  denomination. 

3.  Write  the  last  quotient  and  the  several  remainders  in  their 
order,  from  the  highest  denomination  to  the  lowest,  for  the 
required  result, 

PJtOBLEMS. 

29.  How  many  quart  cui)fuls  in  5  gal.  3  qt.  of  milk  ? 

30.  Reduce  4,879  far.  to  higher  denominations.    £5  1  s.  7.75  d. 

31.  A  10-qt.  pail  holds  how  many  gills  ?     How  many  gallons  ? 
33.  Reduce  167,824  qr.  to  higher  denominations.  £17 Jf  16  s.  4d. 

33.  How  many  times  can  a  2-quart  measure  be  filled  from  a  keg 
which  contains  7.5  gallons  of  vinegar  ? 

Solve  the  above  problem  in  4  different  ways. 

34.  Reduce  .6875  of  a  gal.  to  gills.  23  gi. 

35.  6784.8  d.  are  how  many  pounds  ?    £28.27,  or  £28  5  s.  4.8  d. 

36.  One  week  a  woman  who  kept  a  fruit  stand  sold  19.5  bushels 
of  peanuts  by  the  half-pint  measure.  How  many  measurefiils  did 
she  sell  ? 

37.  Reduce  3  qt.  to  the  decimal  of  a  bushel.  .09375  Tm. 

38.  One  day  a  hostler  at  a  hotel  stable  fed  out  129  half-peck 
measures  of  oats.     How  many  bushels  did  he  feed  out  ? 

39.  In  The  New  Dominion  3,287  cents  are  how  many  dollars, 
shillings,  and  cents  ?  $32  J/s.  7  cts. 

40.  7.3125  s.  are  how  many  farthings  ?  351. 

41.  A  lady  made  3  gal.  2  qt.  of  strawberry  wine,  which  she  put 
into  pint  bottles.    How  many  bottles  of  wine  had  she  ? 

42.  A  housekeeper  filled  549  quart  cans  with  cherries.  How 
many  cherries  did  she  use  ?  17  hu.  5  qt. 


124 


COMPOUND    NUMBERS. 


43.  If  I  measure  3  bu.  1  pk.  5  qt.  of  walnuts  in  a  quart  measure, 
bow  many  times  v/ill  I  fill  the  measure  ? 

44.  Reduce  2  liM.  1  bbl.  15.25  gal.  to  quarts. 

45.  How  many  busliels  of  potatoes  in  12,250  quarts? 

46.  From  a  peach  orchard  1500  bu.  2  pk.  of  peaches  were  sold, 
at  an  average  of  75  cents  a  basket.  How  much  was  received  for 
them  ?  $2250.75. 

Table  VI, 

Linear  Measure, 

226.  The  table  of  linear 
or  line  measure  consists 
of  the  denominations 
miles,  rods,  yards,  feet, 
and  inches.  These  de- 
nominations are  used  in 
measuring  distances,  and 
also  the  dimensions  of 
things,  as  their  length, 
width,  thickness,  height, 
and  dejjth. 

12  in.  (inches)  are  1  ft.  (foot.) 

3  ft.  "1  yd.  (yard.) 

5.5  yd.  "     1  rd.  (rod.) 

320  rd.  "     1  mi.  (mile.) 

Scale.— Ascending,  12,  3,  5.5,  320 


1  mi.  is  320  rd. 
1  rd.  "      5.5  yd. 
1  yd.  "      3  ft. 
1ft.    "     12  in. 
descending,  320,  5.5,  3,  12. 


Topographical,  civil,  and  military  engineers  express  dis- 
tances less  than  a  mile  by  yards  or  feet. 

1760  yd.,  or  5280  ft.  are  1  mi. 

In  measuring  goods  sold  by  the  linear  yard,  the  yard 
is  divided  into  quarters,  eighths,  and  sixteenths. 
2.25  in.  are  1  sixteenth. 

2       sixteenths,  or  4.5  in.     "   1  eighth. 

2       eighths,  or  9  in.  "   1  qr.  (quarter),  or  1  fourth  of  a  yard. 

4       qr.  "    1  yd. 


NOTATION    AND    REDUCTIONS.  125 

Mariners  use  the  following  denominations  : 
G  feet  are  1  fathom,  in  measuring  depths  at  sea. 
120  fathoms  are  1  cable's  length,  for  short  distances. 
1  nautical  mile,  or  knot,  is  1.15  common  or  English  miles. 
1        "       league  is  3  nautical  miles,  or  3.45  English  miles. 

In  geographical  and  astronomical  calculations, 

1  geographic  mi.         is   1.15  statute  mi. ; 

3         "  "  are  1  1.  (league.) 

60  "  "  or  )    "    1  deg.  (degree)  of  latitude,  or  of 

69.16  statute     "        \  longitude  on  the  equator. 

Notes.— 1.  The  knot  is  used  in  measuring  the  speed  of  vessels. 

2.  The  nautical  mile  (or  knot)  and  league  arc  the  same  as  the  geographic 
mile  and  league. 

3.  The  length  of  a  degree  of  latitude  is  not  quite  uniform.  69.16  miles  is 
the  average  length,  and  is  th<3  one  adopted  by  the  U.  S.  Coast  Survey. 

4.  In  measuring  the  height  of  horses,  4  inches  are  1  hand,  the  measure 
being  taken  directly  over  the  fore  shoulder. 

5.  In  clock-making,  6  points  are  1  line,  and  12  lines  are  1  inch. 

6.  In  measuring  the  length  of  the  foot,  3  barleycorns,  or  sizes,  are  1  inch. 

7.  The  sacred  cubit,  mentioned  in  the  Bible,  is  21.888  inches. 

8.  The  old  road  measures,  40  rd.  are  1  fur.  (furlong),  and  8  fur.  are  1  mi., 
are  now  but  little  used. 

Table  VII,— Square  3Ieasure* 

227i  The  table  of  square  or  surface  measure  consists  of 
the  denominations  square  miles,  acres,  square  rods,  square 
yards,  square  feet,  and  square  inches.  These  denomina- 
tions are  used  in  computing  the  area  of  land,  flooring,  plas- 
tering, and  other  surfaces.     Sec  Manual, 

144  sq.  in.  (square  in.)  are  1  sq.  ft. 
9  sq.  ft.  "    1  sq.  yd. 

30.25  sq.  yd.  "    1  sq.  rd. 

IGO  sq.  rd.  "    1  A.  (acre.) 

C40  A.  "    1  sq.  mi. 

Scale.— Ascending,  144,  9,  30.25,  100,  640;  descending,  640, 
160,  30.25,  9,  144. 


1  sq.  mi.  is  640  A. 

1  A.  "  160  sq.  rd. 

1  sq.  rd.  "     30.25  sq.  yd. 

1  sq.  yd.  "       9  sq.  ft. 

1  sq.  ft.  "  144  sq.  in. 


126  COMPOUND     NUMBERS. 

Builders  use  the  following  units  in  estimating  tlieir  work  : 

A  shingle  is  4  inches  wide. 

G  shingles,  laid  6  inches  to  the  weather,  cover  1  sq.  ft. 
100  sq.  ft.  are  1  square  of  roofing  or  flooring. 
100  lath,  or  1  bunch,  cover  5  sq.  yd.  of  surface. 

Notes.— 1.  Glazing  and  stone-cutting  arc  estimated  by  the  square  foot. 

3.  Painting,  plastering,  paper-hanging,  ceiling,  and  paving  are  estimated 
by  the  square  yard. 

3.  Brick-laying  is  estimated  by  the  sq.  yd.,  or  the  square  of  100  sq.  ft. 
In  either  case  the  work  is  understood  to  be  12  in.  or  1.5  bricks  thick. 
Brick-laying  is  also  estimated  by  the  thousand  bricks. 

Table  VIII,— Cubic  Measure. 

228»  The  table  of  cubic  or  solid  measure  consists  of  the 
denominations  cubic  inches,  cubic  feet,  and  cubic  yards. 
These  denominations  are  used  in  computing  the  solidity  of 
timber,  stone,  portions  of  earth,  and  many  other  articles ; 
and  in  estimating  the  capacity  of  bins,  boxes,  etc.,  when 
their  dimensions    are   given  in   denominations   of  linear 

measure.        see  Manual. 

1728  cu.  in.  (cubic  in.)  are  1  cu.  ft.     I     1  cu.  yd.  is      27  cu.  ft. 
27  cu.  ft.  "    1  cu.  yd.  |     1  cu.  ft.    "  1728  cu.  in. 

Scale.— Ascending,  1728,  27  ;  descending,  27,  1728. 

Notes. — 1.  On  public  works  a  cubic  yard  of  earth  is  a  standard  load. 

2.  Railroad  and  transportation  companies  estimate  light  freight  by  the 
space  it  occupies,  in  cubic  feet ;  and  heavy  freight,  by  actual  weight. 

3.  In  estimating  the  tonnage  of  ships  and  other  vessels,  100  cu.  ft.  of 
space  are  1  T.  of  shipping. 

4.  A  perch  of  stone  or  of  masonry  is  16.5  ft.  long,  1  ft.  high,  and  1  ft. 
thick,  or  16.5  cu.  ft. 

5.  A  common  brick  is  8  X  4  X  2  inches,  or  04  cu.  in. 

6.  In  every  cubic  foot  of  bricks,  piled  solid,  are  27  bricks ;  and  in  every 
cubic  foot  of  brick  wall,  laid  in  mortar,  are  22.5  bricks. 

7.  Five  courses  of  bricks  in  the  height  of  a  wall  are  called  a  foot. 

8.  Brick-layers,  masons,  and  joiners  make  a  deduction  of  one  half  the 
space  occupied  by  the  windows  and  doors  in  the  walls  of  buildings. 

9.  In  computing  the  cubic  contents  of  the  walls  of  foundations  and 
buildings,  brick-layers  and  masons  multiply  the  girth  (i.  e.,  the  distance 
round  the  outside  of  the  walls),  height,  and  thickness  together.  By  this 
method  of  measuring,  the  corners  arc  measured  twice. 


NOTATION    AND    REDUCTIONS. 


127 


Table  IX,— Wood  Measure, 


229.  The  table  of 
wood  measure  consists 
of  the  denominations 
cords,  cord  feet,  and  cu- 
bic feet.  These  denomi- 
nations are  chiefly  used 
in  measuring  wood. 
Eough  stone  is  also 
commonly  sold  by  the 
cord.  A  pile  of  wood 
8  feet  long,  4  "feet  wide, 
and  4  feet  high  is  1 
cord  ;  and  1  foot  in 
length  of  such  a  pile  is 
1  cord  foot. 

16  cu.  ft.  are  1  cd.  ft.  (cord  ft.)  ^  ^^     Us  128  cu.  ft., 

8  cd.  ft.,  or  )    a    1  (.^j  (  or     8  cd.  ft. 

128  cu.  ft.      f  '  1  cd.  ft.  is    16  cu.  ft. 

Scale. — Ascending,  16,  8  ;  descending,  8,  16. 

JEXEJRCISES. 

13.  Read  5  mi.  2  yd.  2.7  ft. ;  54  rd.  1.37  yd. 

14.  Read  5  A.  15.25  sq.  yd. ;  3  sq.  mi.  30  sq.  rd.  5.5  sq.  yd. 

15.  Read  11  cu.  yd.  8  cu.  ft.  512  cu.  in. ;  5  cu.  yd.  60  cu.  in. 

16.  Read  45  cd.  7  cd.  ft.  13  cu.  ft ;  31  cd.  5.75  cd.  ft. 

17.  Write  6  miles  85  rods  2  yards  2  feet  8  inches. 

18.  Write  twelve  acres  sixty  square  rods  one  hundred  six  square 
inches. 

19.  Write  16  cubic  yards  18  cubic  feet  350  cubic  inches. 

20.  Write  15  cords  4  cord  feet  13  cubic  feet. 

21.  Write  90  square  rods  14  square  yards  5  square  feet  108  square 
inches. 

22.  Write  15  miles  200  rods ;  11  cubic  yards  1,576  cubic  inches. 


128  COMPOUND    NUMBERS. 


PJB  OBLEMS. 


47.  How  many  feet  are  7  mi.  108  rd.  3  yd.  1  ft.  ?  S8,752. 

48.  Reduce  25  sq.  rd.  3  sq.  yd.  8  sq.  ft.  to  square  inches. 

985,  UO. 

49.  In  7  cu.  yd.  19  cu.  ft.  960  cu.  in.,  liow  many  cubic  inches  ? 

50.  Reduce  18  cd.  5  cd.  ft.  8  cu.  ft.  to  cubic  feet.        2,392  cu.ft. 

51.  Reduce  115,373  in*ches  to  higher  denominations. 

1  mi.  262  rd.  3  yd.  2  ft.  4  in. 

52.  176,169  sq.  in.  are  what  units  of  higher  denominations? 

Jf  sq.  rd.  14  sq.  yd.  8  sq.  ft.  57  sq.  in. 

53.  Reduce  1,001,100,100  cu,  in.  to  higher  denominations. 

21,457  cu.  yd.  1  at.  ft.  580  cu.  in. 

54.  Change  12,875  cu.  ft.  of  wood  to  cords. 

55.  How  many  planks,  averaging  1  ffc.  wide,  will  be  required  for 
a  plank  road  7  mi.  284  rd.  long?  41,646. 

56.  How  many  1-inch  blocks  will  be  required  to  make  a  pile 
that  shall  contain  23  cu.  yd.  18  cu.  ft.  ? 

57.  A  farmer  j)lanted  1  hill  of  com  on  every  square  yard  of 
ground  in  a  field  of  13  A.  96  sq.  rd.  How  many  hills  did  he 
plant?  65,824. 

58.  How  many  loads  must  a  teamster  draw,  to  move  131  cd.  of 
stone,  if  he  draws  1  cd.  ft.  at  a  load  ? 

59.  2  mi.  125  rd.  1.5  ft.  are  how  many  feet  ? 

60.  Reduce  126,720  in.  to  miles. 

61.  How  many  acres  in  a  tract  of  land  6  miles  square  ?  23,040. 
03.  In  4,305,780  sq.  yd.  there  are  how  many  square  miles  ? 

03.  How  much  wood  in  a  pile  160  ft.  long,  4  ft.  wide,  and  9  feet 
high? 

64.  25  cd.  7  cd.  ft.  12.75  cu.  ft.  are  how  many  cubic  feet  ? 

65.  .9  of  a  foot  are  how  many  inches  ?  10.8. 

66.  Reduce  100.8  sq.  rd.  to  the  decimal  of  an  acre.  .63  A. 

67.  .0015  mi.  is  what  decimal  of  a  rod  ?  .48. 

68.  How  many  cubic  yards  are  468,018  cu.  in.  ?  10.03125. 
09.  How  many  acres  in  a  field  125  rd.  long  and  80  rd.  wide  ? 


NOTATION    AND    REDUCTIONS.  129 

70.  One  year  my  potato  crop  yielded  1  bushel  to  the  square  rod, 
and  the  total  yield  was  1,145  bushels.  How  much  land  did  I  plant 
to  potatoes  ?  7  A.  25  sq.  rd. 

71.  A  dealer  in  real  estate  owns  5  rectangular  lots  of  land  of 
1  acre  each ;  and  the  fronts  of  the  lots,  or  their  widths  on  the 
street,  are  3  rd.,  4  rd.,  5  rd.,  8'  rd.,  and  10  rd.  What  are  their 
respective  depths  ? 

72.  A  gentleman  used  5,560  tiles,  each  1  ft.  long,  in  under- 
draining  his  land.     How  much  tile  drain  did  he  put  down  ? 

73.  If  Mississippi  River  deposits  1  inch  of  sediment  at  the  bot- 
tom of  the  Gulf  of  Mexico  each  year,  how  much  will  it  raise  the 
bottom  of  the  gulf  in  1000  years  ?  27  yd.  2  ft.  4  in. 

74.  How  many  cubic  yards  of  earth  must  be  removed,  in  digging 
a  cellar  53  ft.  x  38  ft.  x  8  ft.  ? 

75.  How  many  cords  of  rough  stone  in  a  pile  47  ft.  x  14  ft.  x 
5.5  ft.  ? 

76.  Hovf  many  acres  in'  a  county  of  30  townships  each  6  miles 
square  ? 

77.  How  many  square  inches  are  150  sq.  rd.  95  sq.  in.  ? 

78.  5  mi.  109  rd.  4  yd.  =  how  many  feet  ?  28210.5. 

79.  If  you  measure  only  the  length  and  height  of  a  pile  of  4-foot 
wood,  how  many  feet  of  surface  measure  will  you  allow  for  1  cord  ? 
How  many  cords  of  such  wood  are  there  in  a  pile  75  ft.  long  and  7 
ft.  high  ?  16  cd.  3  cd.  ft.  k  cu.  ft. 

80.  If  the  pile  is  4  feet  high,  how  many  feet  of  running  measure 
(that  is,  length)  will  be  a  cord  ?  How  many  cords  are  there  in  such 
a  pile  185.6  ft.  in  length  ? 

81.  How  much  wood  in  a  j)ilc  of  8-foot  wood  87.5  ft.  long  and 
5.5  ft.  high  ? 

83.  How  many  rods  of  fence  will  be  required  to  inclose  a  tract 
of  land  3  mi.  45  rd.  long  and  335  rd.  wide  ?  1,820. 

83.  A  company  of  immigrants  purchased  a  tract  of  western  land 
4.8  mi.  X  1.75  mi.  in  extent.     How  many  acres  in  the  tract  ? 

84.  Allowing  that  4  persons  can  stand  on  1  square  yard  of  ground, 
how  many  people  can  stand  in  a  street  15  rd.  long  and  35  ft. 
wide?  3,850. 


130 


COMPOUND    NUMBERS. 


Table  X, — Surveyors'  Measures, 

A  Gunter's  chain 


is  4  rods  or  Qt^  feet  long, 
and  consists  of  100  links, 
each  7.92  inches  long. 
This  chain  is  used  by 
surveyors  in  measur- 
ing the  dimensions  or 
boundary  lines  of  land. 
The  table  of  survey- 
ors' measures  consists  of 
the  denominations  miles, 
chains,  and  Hnks,  used 
in  measuring  boundary 
hues  ;  and  townships, 
square  miles  or  sections, 

square  chains,  poles  or  square  rods,  and  square  links,  used 
in  computing  the  area  of  lands. 

1st.  In  Measuring  Dimensions  :  i 

100  1.  (links)  are  1  eh.  (chain.)         I         1  mi.  is    80  ch. 
80  ch.  "    1  mi.  11  ch.  "  100 1. 

Scale. — Ascending,  100,  80  ;  descending,  80,  100. 


2d.  In  Computing  Areas  : 


625  sq.  1.  are  1  P.  (pole)  or  sq.  rd. 

16  P.         "1  sq.  ch. 

10  sq.  ch.  "    1  A. 
640  A.        "    1  sq.  mi.  or  Sec.  (section.) 

36  sq.  mi.  "    1  Tp.  (township.) 

Scale.— Ascending,  625,  16,  10,  640,  36 ;  descending,  36,  640, 
10,  16,  625. 


1  Tp.  is    36  sq.  mi. 

1  sq.  mi  or  Sec.  "  640  A. 
lA.  "    lOsq.ch. 

1  sq.  ch.  "    16  P. 

1  P.  "  625  sq.  1. 


Notes.— 1.  Since  100  links  arc  1  cliain,  1  link  is  .01  of  a  chain,  35  links 
are  .85  of  a  chain,  and  so  on.  Hence,  links  may  be  written  either  as  hun- 
dredths of  a  chain ;  thus  15.44  ch. ;  or  chains  and  links  as  a  compound 
number ;  thus,  15  ch.  44  1. 


NOTATION    AND    REDUCTIONS.  131 

2.  Since  a  chain  is  4  rods  long,  25  links  arc  1  rod.  The  denomination  rod 
is  seldom  used  in  linear  chain  measure. 

3.  Civil  engineers  on  railroads  and  canals  commonly  use  an  engineers' 
chain,  which  consists  of  100  links  each  1  foot  long. 

4.  Sections,  or  square  miles  of  Government  lands,  are  divided  into  8 
equal  parts,  and  each  part,  or  80  acres,  is  often  called  o.  lot  of  land.  Two 
lots,  or  160  acres,  are  called  a  quarter  section.    See  Manual. 

5.  Formerly  40  rods  were  called  a  rood^  and  4  roods  an  acre.  The  denom- 
ination rood,  so  common  in  old  deeds,  mortgages,  and  surveys,  is  now  but 
little  used. 

EXEM  CISES. 

23.  Eead  6  ch.  44  1. ;  3  mi.  53.81  cli. 

24.  Read  3  A.  7  sq.  ch.  12  P.  480  sq.  1. 

25.  Write  1  square  mile  217  acres  6  square  chains  9  poles  145 
square  links. 

26.  Write  45  chains  22  links  both  as  a  compound  number  and  as 
a  denominate  mixed  decimal  number. 

27.  Write  eighty  acres  seven  square  chains  forty-two  square  links. 

JPi2  OBLEMS. 

85.  Reduce  3  mi.  75  ch.  12  1.  to  links.  31,512  I. 

86.  25,000,000  1.  are  how  many  miles  ?  3,125. 

87.  Change  8  A.  14  P.  462.5  sq.  1.  to  square  links. 

88.  Reduce  236,754  square  links  to  higher  denominations. 

89.  The  front  of  a  certain  city  lot  measures  21 1.  3.75  in.  How 
many  inches  front  has  it  ?     How  many  feet  ? 

90.  How  many  acres  in  a  farm  which  is  215  rods  long  by  140 
rods  wide  ?  188.125. 

91.  The  area  of  a  certain  piece  of  land  is  9  sq.  ch.  11.25  P. 
What  is  its  area  in  square  rods  ? 

92.  In  chaining  the  route  for  a  proposed  railroad,  the  engineers 
applied  a  Gunter's  chain  7,254  times.  What  was  the  length  of  the 
route  ?  90.675  mi.,  or  90  mi.  54  ch. 

93.  Walworth  Co.,  Wis.,  consists  of  16  Government  townships. 
How  many  acres  in  the  county  ? 

94.  2.376  in.  is  what  decimal  of  a  chain?  .003. 


132 


COMPOUND    NUMBERS. 


Table  XI. 
Avoirdupois  Weight, 

231.  The  table  of  avoir- 
dupois weight  consists  of 
the  denominations  tons, 
hundred-weigh fc,  pounds, 
and  ounces.  These  de- 
nominations are  used  in 
weighing  most  kinds  of 
produce,  provisions,  gro- 
ceries, metals,  coal,  and 
many  other  articles. 

16  oz.  (ounces)  are  1  lb.     (pound.) 
100  lb.  "1  cwt.  Oiundred-weight.) 

20  cwt.  "    1  T.      (ton.) 

Scale.— Ascending,  16,  100,  20 ;  descending,  20,  100,  16. 

In  wholesale  transactions  in  coal,  iron,  and  iron  ore,  and 
in  invoices  passing  the  United  States  Custom-Houses,  of  all 
English  goods  sold  by  weight, 


IT. 

1  cwt. 
lib. 


is    20  cwt. 
"  100  lb. 
"    16  oz. 


28  lb.   are  1  qr.     (quarter.) 

4  qr.    "    1  cwt. 
20  cwt.  "    1  T. 


1  T.      is  20  cwt. 
1  cwt.  **    4  qr. 
1  qr.      "  28  lb. 


The  ton  of  2,000  lb.  is  commonly  called  the  sliort  t07i,  or 
the  net  ton;  and  the  ton  of  2,240  lb.  (20  cwt.  =  80  qr.  = 
2,240  lb.),  is  called  the  long  ton,  or  the  gross  ton. 

The  following  units  or  denominations  are  in  common  use: 
56  lb.  are  1  bu.  of  salt,  at  the  New  York  State  Salt  Works. 

280  lb.  (5  bu.)   "    1  bar.       "  *'        "        «        "        " 

100  lb.  "    1  cental  of  grain. 

100  1b.  "    1  cask  of  nails  or  raisins. 

196  lb.  "1  bar.  of  flour. 

200  lb.  "    1  bar.  of  beef  or  pork. 

Notes. — 1.  There  is  no  such  thing  as  a  quarter,  of  25  lb.  When  the  quar- 
ter is  named,  it  always  means  28  lb. 

2.  In  theory,  16  dr.  (drams)  arc  1  oz.  But  the  dram  is  neither  used  nor 
recognized  in  business. 


NOTATION    AND    REDUCTIONS. 


133 


3.  Hundred-weight  and  pounds  may  be  read  together  as  pounds,  or  the 
pounds  may  he  read  as  hundredths  of  a  hundred-weight.  Tlius,  4  cwt.  56 
pounds  is  456  lb.,  or  4.56  cwt. ;  5  T.  17  cwt.  9  lb.  is  5  T.  1709  lb.,  or  5  T. 
17.09  cwt. 

232.  The  weight  of  a  bushel  of  the  principal  kinds  of 
grain  and  seeds  has  been  fixed  by  statute  in  many  of  the 
States,  as  shown  in  the  following 

AVOIBDUPOIS  BUSHEL   TABLE. 


50 

3 

48 

'6 

M 

48 

48 

W 
48 

s 

32 

1 
46 

4 
48 

48 

48 

6 
48 

t-3 

48 

48 

6 

1 

48 

5 
46 

a 
47 

> 
46 

45 

Barley,  .  .  . 

48 

Buckwheat, 

40 

45 

40 

50 

52 

52 

46 

42 

42 

52 

50 

50 

48 

42 

48 

46 

42 

42 

Clover  seed, 

60 

60 

60 

60 

60 

60 

60 

64 

60 

60 

60 

60 

60 

Indian  corn. 

52 

56 

56 

52 

56 

56 

56 

56 

56 

56 

56 

52 

54 

56 

58 

56 

56 

56 

56 

56 

56 

Oats,  .  .  .  . 

32 

28 

32 

32 

35 

100  to 
3bu. 

32 

30 

30 

32 

32 

35 

30 

30 

30 

32 

32 

34 

32 

32 

36 

32 

Rye, 

54 

56 

54 

56 

56 

56 

32 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

Timothy  s'd 

45 

45 

45 

45 

45 

44 

46 

Wheat,.  .  . 

60 

56 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60,60 

60 

60 

60|60^ 

60i60 

Table  XII.— Troy  Weight. 

233.  The  table  of  Troy  weight  consists  *of  the  denomina- 
tions pounds,  ounces,  pennyweights,  and  grains.  These 
denominations  are  used  in  weighing  the  precious  metals 
and  jewels,  and  in  philosophical  experiments. 


24  gr.  (grains)  are  1  pwt.  (pennyweight.) 
20  pwt.  "    1  oz. 

12  oz.  "    lib. 


1  lb.     is  12  oz. 
1  oz.     "  20  pwt. 
1  pwt. "  24  gr. 


Scale.— Ascending,  24,  20,  12;  descending,  12,  20,  24. 

Note. — Physicians  in  prescribing,  and  apothecaries  in  mixing,  medicines 
that  are  dry,  divide  the  Troy  pound  according  to  the  following  table  of 

apothecaries'  weight. 

20  gr.  (grains)  are  1  sc.  or  ^  (scruple.) 

3  3  "   1  dr.  or  3  (dram.) 

8  3  "1  oz.  or  3  (ounce.) 

12  I  "   1  lb  (pound.) 

Dry  medicines  are  sold  by  avoirdupois  weight. 


134  COMPOUND    NUMBERS. 

EXEMCISES. 

28.  Read  5  T.  16  cwt. ;  33  T.  1  cwt.  54  lb.  7  oz. ;  2  T.  375.25  lb. 

29.  Read  4  lb.  7  oz.  10  pwt.  20  gr. 

30.  Write  fifty  tons  two  hundred  seven  and  five  tenths  pounds. 

31.  Write  6  pounds  4  ounces  19  pennyweights  12  grains. 

JPJB  OBIjEMS. 

95.  7  T.  15  cwt.  45  lb.  9  oz.  are  how  many  ounces  ?      SJiS,729. 

96.  Change  1,999  oz.  to  hundred-weight.        1  cwt.  2^  lb.  15  oz. 

97.  How  many  grains  are  1  lb.  9  oz.  Troy  weight  ?         10, 080. 

98.  Reduce  5,190  grains  to  ounces.  10  oz.  16  pwt.  6  gr. 

99.  One  day,  9  T.  56  lb.  of  Oswego  corn-starch  were  packed  in 
pound  papers.    How  many  papers  were  put  up  ? 

100.  A  jeweler  made  456  finger  rings,  each  containing  4.25  pwt. 
of  gold.     How  much  gold  did  he  use  ? 

101.  Reduce  .6  gr.  to  the  decimal  of  an  ounce. 

102.  11  oz.  11  pwt.  11  gr.  are  how  many  grains? 

103.  One  year  the  Lebanon  Shakers  put  up  2  T.  16  cwt.  95.75  lb. 
of  garden  seeds  in  papers,  each  containing  .25  of  a  pound.  How 
many  papers  of  seeds  of  this  weight  did  they  put  up  ? 

104.  .00021  T.  is  what  decimal  of  a  pound  ?  .J^2  lb.=6.72  oz. 

105.  One  spring  a  Vermont  farmer  made  161,268  oz.  of  maple 
sugar.    How  many  tons  did  he  make  ?  5  T.  79.25  lb. 

106.  Reduce  .003125  lb.  Troy  to  the  decimal  of  a  grain. 

107.  What  is  the  length  of  a  roll  of  gold  wire  that  weighs  2  lb. 
9  oz.,  if  it  weighs  1  gr.  to  the  foot  ?  3  mi. 

108.  How  many  pounds  of  bluing  will  a  manufacturer  use  in 
putting  up  845,000  1-ounce  boxes  ? 

52812.5  lb.  =  26  T.  812  lb.  8  oz. 

109.  A  wholesale  dealer  bought  2  T.  8  cwt.  of  carpet  tacks  in 
8-oz.  papers.    How  many  papers  of  tacks  did  he  buy  ?       9,600. 

110.  .375  lb.  =  how  many  pennyweights? 

111.  1,250  flat-irons,  weighing  5  lb.  each,  weigh  how  many  tons  ? 

3.125  T.  :=3  T.  250  lb. 


NOTATION    AND    REDUCTIONS. 


135 


Table  XIII,— Time. 

234.  Time  is  a  lim- 
ited portion  of  duration. 

The  table  of  time  con- 
sists of  the  denomina- 
tions centuries,  years, 
months,  weeks,  days, 
hours,  minutes,  and  sec- 
onds. These  denomina- 
tions are  used  in  ex- 
pressing portions  of  time 
or  duration  of  different 
lengths. 

The  day  and  the  year 
are  the  natural  divisions 

of  time,  the  other  denominations,  except  centuries,  being 
parts  of  one  or  the  other  of  these. 

60  sec.  (seconds)  are  1  min.  (minute.) 


60  min.            are  1  h.     (hour.) 

1  century             is  100  yr. 

24  h.                 "   1  da.  (day.) 

I'-p-year          'f^-y^f.' 

7  da.                "   1  wk.  (week.) 

53  wk.  1  da.,  ?  a   1  common  yr. 
or  365  da.  \             (year.) 

lco..on,r.       'f^-y^t: 

53  wk.  3  da.  )   „    .  ,  „^  ^^ 
or  366  da.  J        1  leap-yr. 

Ida.                    "    24  h. 

Ih.                       "     60  min. 

100  yr.                "    1  century. 

1  min.                  "     60  sec. 

Scale.— Ascending,  60,  60,  34, 

365  or  366,  100 ;  descending,  100, 

366  or  365,  34,  60,  60.— The  7  is  ( 

Dmitted  from  the  scale. 

The  length  of  a  solar  year  is  365  da.  5  h.  48  min.  48  sec. 

The  following  years  are  leap-years,  of  366  days  each  : 

I.  Every  centennial  year  that  is  exactly  divisible  by  JfiO  ;  as 
400,  800,  1600  ;  2000,  2400,  2800.     And, 

n.  Every  year  not  a  centennial  year  that  is  divisible  by  Jf. ;  as 
1804,  1808,  1812  ;  1876,  1880,  1892. 

For  explanation  of  the  Calendar,  Bee  Manual. 


136 


COMPOUND    NUMBERS, 


The  year  is  divided  into  12  calendar  months,  and  these 
are  divided  into  4  seasons. 


SEASONS. 

JIONTUS. 

ACBKEVIATIOXS. 

DATS. 

Winter. 

I    1st 
I    2d 

mo. 

January, 

Jan. 

31 

a 

February, 

Feb. 

28  or  29 

(    3d 
]    4th 
^    5th 

a 

March, 

Mar. 

31 

Spring. 

u 

April, 

Apr. 

30 

a 

May, 

31 

(    6th 

u 

June, 

30 

Summer. 

]    7th 

u 

July, 

31 

'    8th 

u 

August, 

Aug. 

31 

C    9th 

a 

September, 

Sept. 

30 

Autumn. 

jlOth 

u 

October, 

Oct. 

31 

'nth 

a 

Kovember, 

Nov. 

30 

Winter. 

12th 

u 

December, 

Dec. 

31 

Notes.— 1.  February  has  28  days  in  a  common  year,  and  29  in  a  leap-year. 
2.  lu  most  business  transactions,  30  days  are  regarded  as  a  montli. 

EXEH  CISES. 

32.  Read  5  yr.  4  mo.  15  da. ;  3  "wk.  5  da.  10  h.  45  min.  30  sec. 

33.  Eead  12  yr.  134  da.  17.35  h. 

34.  Write  4  years  9  months  3  days  30  minutes  15  seconds. 


PJt  OB  JO  J3MS. 

112.  How  many  minutes  in  the  three  spring  months  of  a  common 
year?  132,480. 

113.  IIow  many  hours  from    Independence  day   at  noon,   to 
Christmas  day  at  noon  ?  4j  ^'^^' 

114.  Reduce  573,596  min.  to  higher  denominations. 

56  tch  6  da.  7  7i.  5G  min. 

115.  How  many  seconds  in  a  solar  year? 

116.  How  long  a  time  will  it  take  a  clock  that  ticks  once  every 
second,  to  tick  one  million  times  ?         11  da.  13  h.  46  min.  40  sec. 

117.  My  watch  ticks  4  times  each  second.     How  many  times  wiU 
it  tick  in  a  leap-year  ? 

118.  3  wk.  6  da.  23  h.  30  min.  45  sec.  are  how  many  seconds  ? 


NOTATION    AND    REDUCTIONS.  137 

Table  XIV,— Circular  and  Angular  Pleasure, 

235.  A  Circle  is  a  round,  plane 
surface. 

See  Manual. 

236.  A  Circumference  is 

the  bounding  line  of  a  circle. 

At 

237.  An  Arc  is  any  part  of  the 
circumference  of  a  circle. 

238.  A   diameter  is   the  dis- 
tance across  a  circle  through  its  center,     di" 

239.  A  ^adiics  is  the  distance  from  the  center  to  the 
circumference  of  a  circle. 

Note. — The  radius  of  a  circle  is  always  equal  to  one  half  its  diameter. 

240.  If  the  surface  about  a  point  in  a  plane  be  divided 
into  360  equal  parts  or  spaces,  by  lines  drawn  from  the 
point,  360  equal  angles  will  be  formed,  and  any  one  of  these 
angles  will  be  a  degree.  And  since  an  angle  is  the  differ- 
ence of  dkection  of  two  lines,  or  the  opening  between  two 
lines  that  meet  in  a  point  (see  168),  it  follows  that 

A  degree  is  one  of  the  360  equal  angles  which  will  just 
fill  the  space  about  a  common  point  in  a  plane. 

241.  The  lines  which  form  the  sides  of  these  angles  may 
be  of  any  length ;  and  if  about  their  common  point  of 
meeting  as  a  center,  a  circumference  be  drawn,  cutting  all 
these  lines,  it  will  be  divided  into  360  equal  parts,  and  one 
of  these  parts  will  be  the  measure  of  a  degree,  or  of  the 
angle  at  the  center  of  the  circle.     Hence, 

The  Meastire  of  an  cingte  at  the  center  of  a  circle  is 
that  part  of  the  circumference  included  between  the  sides 
of  the  angle. 

242.  If  the  circumference  of  any  circle  be  divided  into 

360  equal  parts,  each  of  these  parts  is  also  called  a  degree. 

Note.— Since  circles  may  be  great  or  small,  the  degrees  in  their  circum- 
ferences will  he  correspondingly  great  or  small.  An  angle  of  1  degree  is 
constant ;  while  the  measure  of  the  angle,  or  1  degree  in  a  circumference, 
varies  with  every  change  in  the  dimensions  of  the  circle. 


138 


COMPOUND    NUMBERS. 


243.  The  table  of  cir- 
cular and  angular  meas- 
ure consists  of  the 
denominations  circles, 
degrees,  minutes,  and 
seconds.  These  denomi- 
nations are  used 

By  Surveyors^  in  de- 
termining the  directions 
or  bearings  of  land 
boundaries  and  other 
lines  ; 

By  Navigators,  in  de- 
termining the  position 
of  vessels  at  sea  ;  and 

By  Geographers  and  Astronomers,  in  determining  latitude, 
longitude,  and  the  position  and  motion  of  the  heavenly 
bodies  ;  and  in  computing  difference  of  time. 

60"  (seconds)  are  1'  (minute.)  1  C.  is  360° 

60'  "  1°    (degree.)  1°      "     60' 

360°  "  IC.  (circumference.)        1'      "     60" 

Scale. — Ascending,  60,  60,  360 ;  descending,  360,  00,  60. 

Notes.— 1.  A  right  angle  or  a  quadrant  is  an  angle  of  90°,  and  is  always 
included  between  two  lines  perpendicular  to  each  other ;  its  measure  is  one 
fourth  of  a  circumference.  Hence,  we  say  4  right  angles  or  quadrants  are 
1  circumference. 

2.  Navigators  call  one  sixth  of  a  circumference  a  sextant.  Hence,  in  nav- 
igation, 60°  are  1  sextant ;  and  6  sextants  are  1  circumference. 

3.  Astronomers  divide  the  zodiac,  or  the  snn's  apparent  path  in  the  heav- 
ens, into  13  equal  parts,  of  30°  each  (for  the  12  months  of  the  year),  which 
they  call  signs.    Hence,  in  astronomical  calculations, 

30°  are  1  S.  (sign),  and  12  S.  are  1  great  circle  of  the  heavens. 

4.  A  minute  of  the  circumference  of  the  earth  is  1  geographic  mile,  which 
is  1.15  English  miles,  or  1  mi.  48  rd.      See  Manual. 

EXEJRCISES, 

36.  Read  10°  40'  35";  8  S.  25° ;  72°  C  23.75". 

37.  Write  19  degrees  53  minutes  42  seconds. 

38.  Write  one  hundred  five  degrees  twenty-six  geographic  miles. 


NOTATION    AND    REDUCTIONS. 


139 


PR  OBLEMS. 

119.  Reduce  47°  13'  to  seconds.  169,980". 

120.  How  many  degrees  in  59,300''?  16°  28'  20". 
131.  When  a  planet  has  moved  1,426,444"  in  the  heavens,  has  it 

described  more  or  less  than  one  complete  revolution  in  its  orbit  ? 

36°  Uf'  4"  more  than  1  revolution. 

122.  When  the  sun  has  seemed  to  pass  over  8  S.  62.25°  of  the 
zodiac,  how  many  seconds  has  he  seemed  to  move  ? 

123.  How  many  seconds  are  .015°  ?  5^". 

124.  3  quadrants  57°  58'  are  how  many  minutes  ? 

125.  67,875"  =  how  many  degrees  ?  18°  51'  15". 

126.  Reduce  3'  to  the  decimal  of  a  de^rree.  .05°. 


Table  XV. 
Counting, 

244.  The  table  of 
counting  consists  of  the 
denominations  ones, 
dozens,  gross,  and  great 
gross.  These  denomi- 
nations are  used  in 
counting  several  class- 
es of  articles  for  mar- 
ket purposes. 


12  ones  or  things  are  1  doz. 
12  doz.  "    1  gro. 


1  great  gro.  is  12  gro. 
1  gro.  "12  doz. 

1  doz.  "  12  thinsrs. 


(dozen.) 
(gross.) 
12  gro.  "    1  grt.  gro.  (great  gro.) 

Scale. — Ascending  and  descending,  uniformly  12. 

Notes.— 1.  Six  things  of  a  kind  arc  often  called  a  set ;  as  a  set  of  chairs, 
spoons,  forks,  plates,  etc. 

2.  Twenty  things  of  a  kind  are  sometimes  called  a  score;  as  a  score  of 
times,  three  score  of  years,  etc. 


140  COMPOUND    NUMBERS. 


Table  XVI^—JPaper, 


2i5»  The  table  of  paper  consists  of  the  denommations 
bales,  bundles,  reams,  quires,  and  sheets.  These  denomi- 
nations are  used  in  the  paper  trade. 


24  sheets    are  1  quire. 

30  quires     "    1  rm.  (ream.) 

2  rm.         "    1  bundle. 

5  bundles  "    1  bale. 


1  bale      is    5  bundles. 
1  bundle  "    2  rm. 
1  rm.        "  20  quires. 
1  quire     "  24  sheets. 


Scale. — Ascending,  24,  20,  2,  5 ;  descending,  5,  2,  20,  24. 

Note.— Paper  is  bought  at  wholesale  by  the  bale,  bundle,  and  ream ;  and 
at  retail  by  the  ream,  quire,  and  sheet. 

Table  XVII —Copying, 

246.  Lawyers'  clerks  and  copyists  in  public  offices  are 
often  paid  by  the  folio  for  making  copies  of  legal  papers, 
records,  and  documents. 

72  words  are  1  folio,  or  sheet  of  common  law. 
90      "       "    1     "      "      "      "  chancery. 

Table  XVIII.—BooJcs. 

247.  This  table  consists  of  the  terms  used  to  indicate  the 
number  of  leaves  of  a  book  made  from  one  sheet  of  paper. 

When  a  sheet  is  The  book  is 

folded  into  called 

2  leaves,  a  folio, 

4      "  a  quarto  or  4to, 

8      "  -an  octavo  or  8vo, 

12      "  a  duodecimo  or  12mo, 

16      "  a  16mo, 

18      "  an  18mo, 

24      "  a  24mo, 

32      "  a  32mo, 

64      "  a  64mo, 

Note.— A  sheet  of  medium  size  print  paper  is  23  X  36,  24  X  37.5,  or  25  X 
88  inches.    These  are  the  sizes  commonly  used  for  printed  books. 


And  1  sheet  of 

paper 

•  makes 

4  pp. 

(pages.) 

8   " 

16   " 

24   " 

32   " 

36   " 

48   " 

64   " 

128   " 

NOTATION    AND    REDUCTIONS.  141 

i:XEIt  CISES. 

39.  Read  5  grt.  gro.  11  gro.  4  doz. ;  6  gro.  5  doz.  8  steel  pens. 

40.  Write  five  rm.  fifteen  quires  eleven  sheets. 

41.  Write  19  gross  7  dozen  ;  8  great  gross  7.75  dozen. 

JPJB  OBLEMS. 

127.  How  many  gross  are  3,156  buttons  ?  H  gro.  11  doz.  8. 

128.  Hovs^  many  crayons  are  there  in  25  boxes,  each  containing  1 
gross?  8,600.  . 

1 29.  How  many  gross  of  screws  will  a  joiner  use  in  the  26  work- 
ing-days of  a  month,  if  he  uses  56  screws  j^er  day  ?     10  gro.  1  doz.  J^. 

130.  On  inventorying  his  goods,  a  hardware  merchant  finds  that 
he  has  7  gro.  8.5  doz.  wardrobe  hooks  on  hand.  What  number  of 
hooks  has  he  ?  1, 110. 

131.  One  day  a  paper  dealer  sold  6  bales  1  bundle  4  rm.  of      S" 
manilla  wrapping  pajDer.     How  many  sheets  of  paper  did  he  sell  ?  3  /  V^ 

132.  7,260  sheets  of  foolscap  x)aper  are  how  many  reams  ? 

15  rm.  2.5  quires. 

133.  How  many  sheets  of  print  paper  in  a  12mo  book  of  456  pp.  ? 

134.  One  month  a  lady  copied  648.5  common-law  folios  for  a 
lawyer,  at  $.10  per  folio.    How  much  did  she  receive  ? 

135.  A  stationer  has  0  grt.  gro.  11  doz.  lead  pencils.  How  many 
pencils  has  he  ? 

136.  One  winter  a  wood  turner  manufactured  56,870  clothes- 
pins, which  he  packed  in  boxes  of  1  great  gross  each.  How  many 
boxes  had  he  ?  10  gro.  11  doz.  2  pins  more  than  32  boxes. 

137.  A  printed  case  in  the  Supreme  Court  (or  Chancery)  con- 
tained 456,120  words.  How  much  was  the  printer's  bill,  at  $.12|- 
per  folio?  $633.50.  "^ 

138.  How  many  reams  of  print  paper  will  be  required  to  supply 
3,250  subscribers  with  a  weekly  newspaper  one  year  ? 

352  rm.  1  quire  16  sheets. 

139.  A  village  grocer  shipped  5,160  eggs  to  the  city,  in  5  barrels. 
How  many  eggs  did  he  pack  in  a  barrel  ?  86  doz.  p^^  \ 


\«' 


142  COMPOUND    NUMBERS. 

140.  A  bookseller's  stock  of  steel  pens  consists  of  7  packages  of 
1  doz.  boxes  each,  9  boxes  of  a  broken  package,  and  5  doz.  8  pens 
of  an  opened  box.  How  many  pens  lias  he,  each  full  box  contain- 
ing 1  gross?  13,460. 

141.  A  publisher  issued  an  edition  of  5,000  copies  of  an  18mo 
book  of  316  pp.  How  much  paper  did  he  use,  allowing  1  quire  to 
each  ream  for  waste  ?  65  rm.  15  quires  19  sheets. 

Table  XIX. 

Government  Standard  Units  of  Measures  and  Weights, 

248i  The  standard  units  of  the  money,  measures,  and 
weights  now  in  use,  as  adopted  by  the  United  States  Gov- 
ernment in  the  year  1834,  and  from  which  the  other  denom- 
inations in  the  respective  tables  are  determined,  are — 

TABLES.  rNITS.  VALUES. 

United  States  Money,  Dollar,  .900  silver,  .100  alloy. 

Lines,  Surfaces,  and  Solids,  Yard,  3  feet,  or  36  inches. 

Liquid  Measure,  Gallon,  231  cubic  inches. 

Dry  Measure,  Bushel,  2150.43  cubic  inches. 

TroyAVcight,  Pound,  5,760  grains. 

Avoirdupois  Weight,  Pound,  7,000  Troy  grains. 

See  ManuaL 

Notes.— 1.  The  yard  in  use  at  the  Custom-Houses  is  divided  decimally 
into  tenths  and  hundredths. 

2.  For  ordinary  purposes,  2150.4  cu.  in.  are  called  a  bushel. 

3.  In  the  actual  standard  weights  used  by  the       1         oz.  =  480       gr. 
General  Government,  the  Troy  ounce  is  divided         .1       "    -    48       ||_ 
decimally  into  tenths,  hundredths,  thousandths,  .01     ||    =     4.8     ^^ 
and  ten-thousandths.    The  values  of  these  divis-         .001  ^'^'    =       .48   ^^ 
ions  are  shown  in  the  margin  of  this  note.                     .0001"    -       .048 

Table  XX.— Comparative  Values. 
249.  I.  Of  Measures  of  Capacity. 

DENOMINATIONS.  LIQUID   MEASURE.  DRY   MEASURE. 

1  gal.,        231        cu.  in.,       268.8  cu.  in.  (.5  pk.) 
1  qt,  57.75   "     "  67.2  "     " 

ll)t.,  28.875"     "  33.6  "     " 


NOTATION    AND    REDUCTIONS.  143 

n.  Of  Weights. 

DKN0MINATI0N8.  TROY  -WEIGHT.  AVOIEDUPOIS  WEIGHT. 

lib.,  5,760  gr.,  7,000    gr. 

1  oz.,  480  "  437.5  " 

Notes.— 1.  Multiplying  the  number  of  cubic  inches  231         268.8 

in  a  liquid  gallon  by  7,  and  the  number  in  a  dry  gallon       7      6 

by  6,  we  find  that  7  liquid  gallons  contain  4,3  cubic  _^g_^/^       1612.8 
inches  more  than  6  dry  gallons.    Hence,  in  ordinary 

computations,   it    is    sufficiently  accurate   to    estimate  7   liquid   gal.  = 
6  dry  gal. 

3.  Multiplying  the  number  of  grains  in  a  pound  Troy  by  175,  and  the 
number  in  a  pound  avoirdupois  by  144,  we  have 

5,760  X  175  =  7,000  x  144    Hence, 
175  pounds  Troy  =  144  pounds  avoirdupois. 

PB^OBZEMS, 

143.  How  many  more  cubic  inches  in  568.5  gallons  dry  measure, 
than  in  the  same  number  of  gallons  liquid  measure  ?         21489.3. 

143.  If  you  dip  33  quarts  of  water  into  a  tub  that  will  hold  33 
quarts  of  wheat,  how  much  will  the  tub  lack  of  being  full  ? 

The  above  problem  can  be  solved  in  at  least  five  different  ways. 

144.  What  is  the  difference  in  the  weight  of  43.375  pounds  of 
lead  and  43.375  pounds  of  silver  ? 

145.  100  pounds  avoirdupois  are  how  many  pounds  Troy  ? 

146.  A  brewer  has  a  vat  that  will  hold  5,000  gallons  of  beer. 
How  many  bushels  of  barley  will  it  hold  ? 

147.  A  grocer  buys  3  bu.  of  chestnuts,  at  $5.00  a  bushel,  wooden 
measure,  and  retails  them  at  $.30  a  quart,  tin  measure.  How  much 
does  he  gain  ?  $7.40. 

NoTK  3. — Among  many  retailers,  dry  measure  is  called  Wooden  Measure, 
and  liquid  measure.  Tin  Measure. 

148.  An  express  agent  in  "Washington  collected  charges  at  the 
Mint,  for  transporting  563  pounds,  commercial  weight,  of  silver 
from  California.  How  many  Mint  pounds  of  silver  were  trans- 
ported ?  682  lb.  11  oz.  16  pwt.  16  gr. 


144 


COMPOUND    NUMBERS. 


250.    The  Metric  System  of  Weights  and  Measures, 

In  the  year  1866,  the  Congress  of  the  United  States 
passed  a  bill  authorizing  the  use  of  a  new  system  of 
weights  and  measures.  In  this  system  the  principal  de- 
nomination is  the  Metre,  from  which  all  the  other  denom- 
inations in  all  the  tables  are  derived.  Hence,  this  system  is 
called  the  Metric  System. 

The  principal  denomination  for  the  Measure  of  Surface 
is  the  Are  ;  for  the  Measure  of  Capacity,  the  Litre  ;  and  for 

Weight,  the  Gram.        see  Manual. 

The  lower  denominations  in  each  table  are  tenths,  hun- 
dredths, or  thousandths  of  these  ;  and  their  names  are 
formed  by  prefixing  deci,  centi,  or  milli  to  the  name  of  the 
principal  denomination. 

The  higher  denominations  are  10,  100,  1,000,  or  10,000 
times  the  principal  denomination  of  any  table  ;  and  theii* 
names  are  formed  by  prefixing  deka,  hecto,  kilo,  or  myria  to 
the  name  of  that  principal  denomination. 

TABLE  OF  DENOMINATIONS  AND  THEIR  RELATIVE  VALUES. 


PREFIXES  FOR 
LOWER  DENOMINATIONS. 

Milli  (mill-ee)  .001  of 
Centi  (sent-ee)  .01  of 
Bed  (des-ee)   .1      of 


NAMES  OF 

PRINCIPAL 

DENOMINATIONS. 

Metre  (mee-ter) 
Are      (are) 
Litre   (li-ter) 
Gram 


PREFIXES   FOR 
UIGUER  DENOMINATIONS. 

Deka    (dek-a)  10 

Hecto   (hec-to)  100 

Kilo     (kill-o)  1,000 

Myria  (inir-e-a)  10,000 


MEASURES    or   LENGTH. 


10  millimetres  are  1  centimetre. 


10  centimetres  " 
10  decimetres  " 
10  metres  " 

10  dekametres  " 
10  hectometres  " 
10  kilometres     " 


1  decimetre. 
1  metre. 
1  dekametre. 
1  hectometre. 
1  kilometre. 
1  myriametre. 


1  millimetre 
1  centimetre 
1  decimetre 
I  METRE 
1  dekametre 
1  hectometre 
1  kilometre 
1  myriametre 


.001  metre. 

.01    metre. 

.1      metre. 

39.37  i7ichcs. 

10  metres. 

100  metres. 

1,000  metres. 

10,000  metres. 


off  Tl^ 

NOTATION    AND    EE  D  U  C  T  10  N  S„C*9  .   IW    ^ifl 


MEASURES   OF   SUEFACE. 

1  centare  is    .01  are. 

I  ARC       u  i  1^^  s<l-  metres,  or 

1  hectare  "    100  ares. 


100  centares  are  1  are. 
100  ares  "    1  hectare. 


MEASURES   OF    CAPACITY. 


10  millilitres  are  1  centilitre. 
10  centilitres   "     1  decilitre. 
10  decilitres     "     1  litre. 

1  dekalitre. 
1  hectolitre. 


10  litres  " 

10  dekalitres   " 


lOhectoUtres  "  |  ^  g^^rl!^'"' ^' 


1  millilitre  is  .001  litre. 
1  centilitre  "  .01  litre. 
1  decilitre    "   .1      litre. 

r  1  cu.  decimetre,  or 
I  LITRE       'M  j    .908    dryqt. 

[  \  1.0567  liquid  qt. 
1  dekalitre  "       10  litres, 
1  hectolitre  "     100  litres. 
1  kilolitre 
or  stere 


I" 


1000  litres. 


WEiaHT. 


10  milligrams  are 

10  centigrams  " 

10  decigrams  " 

10  grams  " 

10  dekagrams  " 

10  hectograms  " 

10  kilograms  )  <; 

or  kilos      \ 
10  myriagrams 

10  quintals 


1  centigram. 
1  decigram. 
1  gram. 
1  dekagram. 
1  hectogram. 
1  kilogram. 

1  myriagram. 

1  quintal. 
1  millier, 
or  tonneau. 


1  milligram 
1  centigram  ' 
1  decigram  * 
I  GRAM 
1  dekagram  * 
1  hectogram  ' 
1  kilogram  or^ 
I  KILO 

1  myriagram  ' 
1  quintal  ' 
1  millier  ' 


13 


.001  gram. 
.01  gram. 
.1  gram. 
15.Ji32  grains. 

10  grams. 

100  grams. 

1000  grams,  or 

2.20J^6  pounds. 

10  kilos. 

100  kilos. 

1,000  kilos. 


Note. — The  weights  and  measures  of  this  system  have  not  yet  come  mto 
use  in  this  country.  They  are  in  general  use  in  France,  Belgium,  Spain, 
and  Portugal ;  and  their  use  has  been  legalized  by  Great  Britain,  Italy, 
Norway,  Sweden,  Greece,  Mexico,  and  most  of  the  South  American  gov- 
ernments. 


1  wJc.  Jf  da, 
5        0 
2 
9         J, 
3         0 

LUTJIOJN. 

18  h.  Ji5  min, 
U     SO 
20      25 
11 
9        8 

19  wJc.  6  da. 

1  h.  Jf8  min. 

146  COMPOUND    NUMBERS. 

SECTION  III. 

251.  Ex.  What  is  the  sum  of  1  wk.  4  da.  18  h.  45  min., 
5  wk.  14  h.  30  min.,  2  da.  20  li.  25  min.,  9  wk.  4  da.  11  h., 
and  3  wk.  9  h.  8  min.  ? 

Explanation. — Since  only  like 
orders  of  units  in  different  num- 
bers can  be  added  (see  39,  II.), 
we  write  the  numbers  with  like 
denominations  —  or  orders  of 
units  —  in  the  same  columns. 
Then,  commencing  at  the  right, 
we  add  the  units  of  each  denomination,  in  order,  from  the 
lowest  to  the  highest.  The  sum  of  the  minutes  is  108,  or 
1  h.  48  min.  We  write  the  48  min.  as  the  minutes  of  the 
sum,  and  add  the  1  h.  with  the  hours  of  the  given  numbers. 
The  sum  of  all  the  hours  is  73,  or  3  da.  1  h.  We  write  the 
1  h.  as  the  hour  of  the  sum,  and  add  the  3  da.  with  the 
days  of  the  given  numbers.  The  sum  of  all  the  days  is  13, 
or  1  wk.  6  da.  We  write  the  6  da.  as  the  days  of  the  sum, 
and  add  the  1  wk.  with  the  weeks  of  the  given  numbers. 
The  sum  of  all  the  weeks  is  19,  which  we  write  as  the  weeks 
of  the  sum.  The  result,  19  wk.  G  da.  1  h.  48  min.,  is  the 
sum  required. 

In  integers  and  decimals,  the  units  of  each  order  are 
added  separately,  and  1  is  added  or  carried  to  the  next 
higher  order,  for  every  10  in  the  sum  of  the  order  added. 
(See  39,  HI.) 

In  compound  numbers  the  units  of  each  denomination— 
or  order — are  added  separately,  and  1  is  added  or  carried 
to  the  next  higher  denomination,  for  as  many  units  in  the 
sum  of  the  denomination  added,  as  equal  1  of  the  next 
higher  denomination.     That  is, 


ADDITION.  147 

I.  In  addition  of  integers  and  decimals^  the  carrying  unit  is 
uniformly  10  ;  and, 

II.  In  addition  of  compound  numbers,  the  carrying  unit  in 
any  denomination  is  that  number  in  the  scale  which  equals  1  of 
the.  next  higher  denomination. 

252.  Upon  these  principles  is  based  the 

^lele  for  A.ddiUon  of  Compound  J^umbers, 

I.  Add  the  units  of  each  denomination  separately,  and  when 
the  sum  is  less  than  a  unit  of  the  next  higher  denomination, 
write  it  in  the  result. 

II.  WJien  the  sum  of  the  units  of  any  denomination  is  equal 
to  one  or  more  units  of  the  next  higher  denomination,  write  the 
excess  in  the  result,  and  add  the  number  of  units  of  the  higher 
denomination  with  the  given  units  of  that  denomination. 

JPJB  OBIjBMS. 

1.  In  four  days  a  gardener  sold  12  bu.  3  pk.,  10  bu.  2  pk.,  8  bu. 
1  pk.,  and  7  bu.  3  pk.  of  peas.     How  many  peas  did  he  sell? 

2.  A  milkman  sells  in  6  successive  days  200  qt.  1  pt.,  220  qt., 
215  qt.  1  pt.,  208  qt.,  199  qt.,  and  187  qt.  1  pt.  of  milk.  How  many 
gallons  does  he  sell  ?  S07  gal.  2  qt.  1  pt. 

(3)  (4)  (5) 

26  lb.  8  oz.  18  pwt.  12  gr.  18°  16'  21.3''  £17    5s.  6d. 

15        6        11             0  25    56  45  32  15     9 

9          0         '22  87   45  39.75  487  00  11  1  far. 

14        3        17             9                16     3  2 


6.  One  winter,  a  wood-cutter  chopped  7  piles  of  wood,  that 
measured  29  cd.  6  cd.  ft.  8  cu.  ft.,  38  cd.  4  cd.  ft.,  31  cd.  2  cd.  ft. 
4  cu.  ft.,  43  cd.  7  cd.  ft.,  21  cd.  3  cd.  ft.  4  cu.  ft.,  34  cd.  7  cd.  ft., 
and  38  cd.  2  cd.  ft.     How  much  wood  did  he  chop  ?  238  cd. 

7.  A  coal  dealer  bought  at  the  Scranton  coal  mines  in  Pennsyl- 
vania 108  T.  13  cwt.  1  qr.  16  lb.  of  stove  coal,  87  T.  7  cwt.  2  qr. 
20  lb.  of  chestnut  coal,  76  T.  19  cwt.  3  qr.  4  lb.  of  large  egg  coal, 
69  T.  3  qr.  of  small  egg  coal,  and  41  T.  19  cwt.  22  lb.  of  lump  coal. 
How  much  coal  of  all  kinds  did  he  buy  ?  384  T.  3  qr.  6  lb. 


148  COMPOUND    NUMBEKS. 

8.  From  a  cask  of  vinegar,  8  gal.  3  qt.  1  pt.  were  drawn  one  day, 
7  gal.  1  qt.  tlie  second,  13  gal.  1  pt.  the  third,  and  9  gal.  3  qt.  1.5 
pt.  the  fourth.     How  much  was  drawn  from  the  cask  ? 

9.  A  dairy-man  makes  515  lb.  8  oz.  of  butter  in  June,  499  lb. 
13  oz.  in  July,  496  lb.  15  oz.  in  August,  and  489  lb.  9  oz.  in  Sep- 
tember.    How  much  butter  does  he  make  in  the  four  months  ? 

10.  How  much  land  in  5  farms  which  contain  335  A.  135.75  sq. 
rd.,  99  A.  18  sq.  rd.,  545  A.  88.35  sq.  rd.,  137  A.  43.5  sq.  rd., 
and  333  A.  143.375  sq.  rd.  ? 

11.  The  cargo  of  a  canal-boat  consisted  of  33  T.  17  cwt.  35  lb.  of 
pig-iron,  18  T.  9  cwt.  48  lb.  of  agricultural  implements,  14  T.  14 
cwt.  of  flour,  and  3  T.  7  cwt.  38  lb.  of  salt.  What  was  the  weight 
of  the  cargo  ?  60  T.  8  cwt.  1  lb. 

13.  The  ceiling  of  a  room  contains  33  sq.  yd.  8  sq.  ft.  94  sq.  in.  of 
plastering,  the  side  walls  33  sq.  yd.  8  sq.  ft.  130  sq.  in.  each,  and  the 
two  end  walls  31  sq.  yd.  130  sq.  in.  each.  How  much  plastering  is 
there  in  the  room  ?  122  sq.  yd.  I42  sq.  in. 

13.  I  bought  meat  as  follows  : 

March  8,   3  lb.  11  oz.  of  steak  ;  3  lb.    9  oz.  of  muttoa 
"     10,    3  lb.    6  oz.  "       "        3  lb.    6  oz.  "       " 
"     13,    1  lb.  15  oz.  "       "        3  lb.  13  oz.  "       " 
How  much  meat  of  each  kind  did  I  buy  ? 

14.  Mr.  Young  has  a  farm  of  six  sides,  which  measure  41  ch.  66  1., 
15  ch.  94  1.,  37  ch.  33  1.,  18  ch.  10  1.,  36  ch.  40  1.,  and  48  ch.  73  1. 
long.    What  length  of  fence  is  required  to  inclose  it  ? 

15.  In  grading  up  a  building  lot,  I  used  the  earth  dug  from  four 
cellars.  The  first  cellar  was  33  x  33  ft.,  the  second  38  x  30  ft., 
the  third  34  x  18  ft.,  and  the  fourth  38  x  18  ft.,  and  each  of  them 
was  4  feet  deep.     How  much  earth  did  I  use  ?   325  cu.  yd.  25  cu.ft. 

16.  What  is  the  sum  of  4  mi.  380 

rd.  3  yd.  3  ft.,  6  mi.  130  rd.  1  yd.  bohttion. 

3  ft.,  1  mi.  96  rd.  5  yd.  1  ft.  ?  ^  r^^  ^80  rd.  2  yd.  2  ft, 

Explanation. — The  sum  of      Q        ISO        1        2 

[  96        5        1 


the  feet  is  5,  or  1  yd.  2  ft.      i_ 

The  sum  of  the  yards  is  9,  S  yd.  2  ft. 


1       6  in. 


or  1  rd.  3.5  yd.    Writing  the 

3  yd.  under  the  column  of     12  mi.  187  rd.  4.  yd.  Oft.  6  in. 


SUBTRACTION.  *  149 

yards,  we  reduce  tlie  .5  yd.  to  units  of  lower  denomina- 
tions. .5  yd.  =  1.5  ft.,  and  .5  ft.  =  6  in.  We  write  tlie  1  ft. 
under  the  2  ft.  already  obtained,  and  the  6  in.  at  the  right 
of  the  1  ft.  Then,  commencing  again  at  the  right,  we  add 
the  inches,  feet,  and  yards  of  the  two  partial  results,  after 
which  we  proceed  with  the  remaining  denominations  of  the 
given  numbers. 

17.  Add  17  mi.  196  rd.  5  yd.  1  ft.,  3  mi.  12  rd.  3  yd.  3  ft.  6  in., 
1  mi.  76  rd.  4  yd.  1  ft.  4  in.,  and  3  mi.  156  rd.  5  yd.  3  ft.  8  in. 

2Ji.  mi.  123  rd.  3  yd. 


SECTION  IV. 

S  ITS  T^A.  CTIOJV. 

253.  Ex.  1.  From  17  yd.  1  ft.  9  in.  take  9  yd.  2  ft.  5  in. 

Explanation. — Since   only  like  orders 
of  units  in  different  numbers  can  be  sub-  solution. 

tracted  the  one  from  the  other  (see  62,       -^^  V^-  ^fi-  ^  ^^• 
II.),   we   write    the    numbers  with  the 


denominations  of  the  subtrahend  under  ^  2/^*  ^ft-  ^  ^^' 
like  denominations — or  orders  of  units 
— of  the  minuend.  Then,  commencing  at  the  right,  we  sub- 
tract the  units  of  each  denomination  of  the  subtrahend 
from  the  like  denomination  of  the  minuend,  in  order,  from 
the  lowest  to  the  highest.  5  in.  from  9  in.  leave  4  in.,  which 
we  write  as  the  inches  of  the  remainder.  Since  we  can  not 
subtract  2  ft.  from  1  ft.,  and  since  the  difference  wiU  not  be 
affected  by  adding  the  same  number  to  both  minuend  and 
subtrahend  (see  52,  III.),  we  add  3  ft.  to  the  1  ft.  of  the 
minuend,  and  1  yd.  (=  3  ft.)  to  the  9  yd.  of  the  subtrahend. 
We  then  subtract  2  ft.  from  4  ft.,  and  10  yd.  from  17  yd., 
writing  the  2  ft.  and  the  7  yd.  as  the  feet  and  yards  of  the 
remainder.  The  result,  7  yd.  2  ft.  4  in.,  is  the  remainder 
required. 


150  COMPOUND    NUMBERS. 

Ex.  2.  From  3  lb.  12  pwt.  take  7  oz.  14  pwt.  9.25  gr. 

Explanation. — In  solving  this 
example,  we  first  add   24   gr.  solution. 

to  the  minuend,   and  1  pwt.      3  Ih,  0  oz,  12 pwt. 

{^  24  gr.)  to  the  14  pwt.  of  the      I J^  0.05  gr, 

subtrahend,  and  subtracting  -  lb,  4  oz,  17  pwt,  U.75  gr. 
9.25  gr.  from  24  gr.,  we  obtain 

14.75  gr.  We  next  add  20  pwt.  to  the  12  pwfc.  of  the  minu- 
end, and  1  oz.  (=  20  pwt.)  to  the  7  oz.  of  the  subtrahend, 
and  subtracting  15  ^vfi.  from  32  pwt.,  we  obtain  17  pwt. 
We  then  add  12  oz.  to  the  minuend,  and  1  lb.  (=  12  oz.) 
to  the  subtrahend,  and  subtracting  8  oz.  from  12  oz.,  we 
obtain  4  oz.  Finally,  subtracting  1  lb.  fi'om  3  lb.,  we  obtain 
2  lb.  The  result,  2  lb.  4  oz.  17  pwt.  14.75  gr.,  is  the  re- 
mainder required.       see  Manual. 

In  integers  and  decimals,  when  the  units  of  any  order  in 
the  subtrahend  are  more  than  those  of  the  Uke  order  in  the 
minuend,  10  units  of  the  same  order  are  added  to  the  min- 
uend, and  1  unit  of  the  next  higher  order  is  added  to  the 
subtrahend. 

In  compound  numbers,  when  the  units  of  any  denomina- 
tion in  the  subtrahend  are  more  than  those  of  the  like 
denomination  in  the  minuend,  as  many  units  of  the  same 
denomination  as  equal  1  unit  of  the  next  higher  denomina- 
tion are  added  to  the  minuend,  and  1  unit  of  the  next 
higher  denomination  is  added  to  the  subtrahend. 

That  is,  whenever  the  units  of  any  order  or  denomination 
in  the  subtrahend  are  more  than  those  of  the  same  order  in 
the  minuend, 

I.  In  integers  and  decimals,  10  units  of  the  same  order  are 
really  added  to  both  terms  ;  and, 

II.  In  compound  numbers,  as  many  units  of  the  same  denom- 
ination are  really  added  to  both  terms  as  equal  1  unit  of  the  next 
higher  denominaiion. 


SUBTRACTION.  tbl 

254.  Upon  these  principles  is  based  the 

^ute  for  Subtraction  of  Comjjoteud  JVurnbers, 

I.  Subtract  the  units  of  each  denomination  separately,  writing 
the  difference  for  the  units  of  the  same  denomination  in  the 
result. 

II.  When  the  units  of  any  denomination  in  the  subtrahend 
are  more  than  those  of  the  like  denomination  in  the  minuend, 
add  to  the  minuend  as  many  units  of  the  same  denomination  as 
equal  1  unit  of  the  next  higher  denomination,  and  to  the  sub- 
trahend 1  unit  of  the  next  higher  denomination. 

PMOBJLEMS  , 

1.  If  my  trees  yield  11  bu.  3  pk.  6  qt.  of  cherries,  and  I  "wish  3 
bu.  1  pk.  4  qt.  for  my  own  use,  how  many  cherries  will  I  have  to 
sell?  81m.2pTc.2qt. 

2.  From  a  bin  containing  433  bu.  3  pk.  of  wheat,  256  bu.  1  pk. 
6  qt.  have  been  taken.     How  much  remains  in  the  bin  ? 

(3)  (4) 

From       17  gal.  2  qt.  1  pt.  2  gi.  5  sq.  mi.  180  A.  4  sq.  ch. 

Subtract    9         3       13  1  576       8 


7  gal.  2  qt.  1  pt.  3  gi.  3  sq.  mi.  243  A.  6  sq.  ch. 

5.  A  grocer  bought  a  jar  of  butter,  weighing  56  pounds.  After 
selling  17  lb.  14  oz.,  how  much  had  he  left  ? 

6.  A  jeweler  used  5  oz.  8  pwt.  15  gr.  of  gold  in  making  8  oz. 

5  pwt.  12  gr.  of  jewelry.     How  much  alloy  was  used  ? 

2  oz.  16  pwt.  21  gr. 

7.  A  load  of  hay  with  the  wagon  weighs  1  T.  9  cwt.  65  lb.,  and  the 
wagon  alone  weighs  11  cwt.  36  lb.  What  is  the  weight  of  the  hay? 

8.  One  morning,  at  a  wood  yard  on  Mississippi  River,  there  were 
on  hand  1,753  cd.  2  cd.  ft.  of  wood,  and  during  the  day  119  cd. 

6  cd.  ft.  were  sold  to  passing  steam-boats.    How  much  wood  was  in 
the  yard  at  night  ?  1, 033  cd.  4  cd.  ft. 

9.  Providence  is  situated  m  longitude  71°  24'  48"  west,  and  San 
Francisco  in  longitude  122°  23'  west.  How  much  farther  west  is 
San  Francisco  than  Providence  ?  ^0°  58'  12". 


152  COMPOUND    NUMBERS. 

10.  From  a  farm  that  contained  213  A.  40  sq.  rd.,  I  sold  98  A.  128 
sq.  rd.     How  much  land  remained  in  the  farm  ?   llJf  A.  72  sq.  rd. 

11.  From  a  cellar  containing  289  cu.  yd.  17  cu.  ft.  of  earth,  175 
cu.  yd.  25  cu.  ft.  were  taken.  How  much  earth  remained  in  the  cellar  ? 

12.  From  a  cask  containing  42  gal.  3  qt.  of  vinegar,  I  sold  15  gal. 
3  qt.  l.G  pt.     How  much  had  I  left  ? 

13.  Two  masons  put  on  533  sq.  yd.  3  sq.  ft.  of  wall,  one  of  them 
plastering  299  sq.  yd.  6  sq.  ft.     How  much  did  the  other  plaster  'i 

14.  A  coal  dealer  having  bought  545  T.  3  qr.  of  coal,  sold  26  T. 
3  qr.  25  lb.     How  much  had  he  left  ? 

15.  A  stationer  bought  30  gro.  4  doz.  lead-pencils,  and  immedi- 
ately afterward  sold  9  gro.  8  doz.  6  of  them.  How  many  pencils 
had  he  left  ?  20  gro.  7  doz.  6. 

16.  An  excavation  45  x  22  x  8  ft.  was  to  be  made,  and  215 
cu.  yd.  28.25  cu.  ft.  of  it  has  been  excavated.  How  much  remains 
to  be  done  ? 

17.  From  17  mi.  4  rd.   2  yd.  solution. 

1  ft.,  subtract  14  mi.  4  yd.  2  ft.  17  mi.  Jf  rd.  2  yd.  1  ft. 

Explanation.— 2  ft.  from  4:      U        0        Jf        2 ^ 

ft.  leave  2  ft.     5  yd.  from  7.5  2  yd.  2  ft. 

yd.  (2  yd.  +  5.5  yd.)  leave  2.5 ^      ^'^n. 

yd.  =  2  yd.  1  ft.  6  in.  We  3  mi.  8  rd.  S  yd.  Oft.  6  in, 
write   the   2  yd.   under    the 

yards  of  the  given  numbers,  and,  since  the  1  ft.  6  in.  be' 
longs  with  the  feet  and  inches  of  the  final  result,  we  add 
them  to  that  part  of  the  result  already  found,  and  obtain 
3  yd.  0  ft.  6  in.     We  then  finish  the  solution  as. already 

taught.        See  Manual. 

18.  From  25  mi.  2  ft.  11  in.,  take  132  rd.  3  yd.  8.75  in. 

2Jf  mi.  187  rd.  3  yd.  8.25  in. 

19.  The  length  of  gas  pipe  in  use  in  a  certain  city  last  year  was 
23  mi.  194  rd.  2  yd.,  and  now  it  is  25  mi.  46  rd.  1  yd.  How  much 
pipe  has  been  laid  during  the  year  ?     1  mi.  171  rd.  ^  yd.  1ft.  6  in. 

20.  What  is  the  difference  between  31  rd.  5  yd.  2  ft.  11  in.  and 
32  rd.  1  ft.  4  in.  ?  i  in. 


SUBTRACTION.  153 

255.  Difference  between  any  two  Dates. 

Ex.  How  many  years,  montlis,  and  days  elapsed  between 
May  21,  1869,  and  Sept.  14,  1871  ? 

Explanation. — Since  the  later  of  solution. 

two  dates  is  always  expressed  by  a      1871  yr.  9  mo.  IJ^  da, 
greater  compound  number  than  the       J^'^P       5         21 
earlier,  we  subtract  the  compound  2  yr.  S  mo.  28  da. 

number  expressing  the  earlier  date 

from  that  expressing  the  later,  writing  the  number  of  the 
year,  month,  and  day  of  each  date  in  order,  as  shown  in 
the  Solution.  Whenever  the  number  of  days  in  the  sub- 
trahend is  greater  than  that  in  the  minuend,  we  call  thirty 
days  a  month.     (See  231,  Note  2). 

rn  OBJLEMS, 

31.  A  note  dated  May  11, 1864,  was  paid  Sept.  25,  1865.  How 
long  did  it  remain  unpaid  ?  1  yr.  Ji.  mo.  IJi,  da. 

22.  The  civil  war  in  the  U.  S.  commenced  April  12,  1861,  and 
closed  May  26,  1865.     "What  length  of  time  did  it  continue  ? 

23.  A  note  given  May  22,  1868,  was  paid  Aug.  10,  1869.  How 
long  did  the  note  run  ? 

24.  George  Washington  died  Dec.  14,  1779,  aged  67  yr.  9  mo.  22 
da.    What  was  the  date  of  his  birth  ?  Feb.  22,  1732. 

25.  How  much  time  has  passed  since  his  death  ? 

26.  Mary  was  13  yr.  8  mo.  12  da.  old  July  15,  1867.  What 
was  the  date  of  her  birth  ?  Nov.  3,  1853. 

27.  A  note  was  given  Aug.  12,  1865,  payable  Feb.  4, 1866.  How 
long  had  it  to  run  ?  5  mo.  22  da. 

28.  Henry  was  bom  Dec.  5,  1852.  How  old  was  he  March  19, 
1868  ?  15  yr.  3  mo.  U  da. 

29.  March  7,  a  mason  contracts  to  build  five  bridge  abutments 
that  shall  contain  856  pch.  8  cu.  ft.  of  stone  work,  and  to  have 
the  work  done  July  1.  May  24,  he  has  finished  594  pch.  12  cu.  ft. 
How  much  work  remains  to  be  done,  and  how  much  time  has  he 
in  which  to  finish  it  ?  261  pch.  12.5  cu.  ft. ;  1  mo.  7  da. 

7* 


SOLUTION. 


154  COMPOUND    NUMBERS. 

SECTION    V. 

Mrz  TUPZICA.  TIOJ\r. 
256i  Ex.  How  mucli  is  7  times  38  bu.  1  pk.  5  qt. 

Explanation. — We  write  the  multi- 
plier under  the  lowest  denomination         S8  hu.  1  pk.  5  qt. 

of  the  multiplicand,  and  multiply  the ^ 

units  of  each  denomination  of  the  mul-  2G8  hu.  3  pk.  3  qt. 
tiphcand  by  the  multiplier,  in  order, 
from  the  lowest  to  the  highest.  Since  the  product  is  always 
of  the  same  kind  or  denomination  as  the  true  multiplicand 
(see  80,  IV.),  7  times  5  qt.  are  35  qt.,  or  4  pk.  3  qt.  We 
write  the  3  qt.  as  the  quarts  of  the  product,  and  reserve 
the  4  pk.  to  be  added  with  the  pecks  of  the  product. 
7  times  1  pk.  are  7  pk.,  and  7  pk.  +4  pk.  =  11  pk.,  or  2  bu. 
3  pk.  We  write  the  3  pk.  as  the  pecks  of  the  product,  and 
reserve  the  2  bu.  to  be  added  with  the  bushels  of  the  prod- 
uct. 7  times  38  bu.  are  266  bu.,  and  266  bu.  +  2  bu.  = 
268  bu.,  which  we  write  as  the  bushels  of  the  product. 
The  result,  268  bu.  3  pk.  3  qt.,  is  the  product  required. 

In  integers  and  decimals,  the  units  of  each  order  are 
multiplied  separately,  and  1  is  added  to  the  product  of  the 
next  higher  order,  for  every  10  in  the  product  of  the  two 
figures  multiplied. 

In  compound  numbers  the  units  of  each  denomination 
are  multiplied  separately,  and  1  is  added  or  carried  to  the 
product  of  the  next  higher  denomination,  for  as  many  units 
in  the  product  of  the  denomination  multipHed,  as  are  equal 
to  1  of  the  next  higher  denomination.     That  is, 

-I.  The  carrying  unit  in  multiplication  of  integers  and  deci- 
mals is  10,  the  same  as  in  addition  ;  And 

II.  Tlie  carrying  unit  in  multiplication  of  compound  num- 
herSjis  determined  from  the  table  or  scale  to  which  the  compound 
number  belongs,  the  same  as  in  addition*     Hence, 


MULTIPLICATION.  155 

257.  ^ute  for  MuUipUcatioji  of  Comj)ou7id  JVumberso 

I.  Multiply  the  units  of  each  denomination  hy  the  whole 
multiplier^  as  in  integers  and  decimals. 

II.  Carry  from  a  lower  to  a  higher  denomination  in  the  pro- 
duct, for  that  number  in  the  table  or  scale  corresponding  to  the 
denomination  multiplied,  as  in  addition  of  compound  numbers. 

Pit  OB  ZJEMS. 

1.  How  much  seed  wheat  will  it  take  to  seed  93  acres,  using  1 
bu.  3  pk.  4  qt.  to  the  acre  ?  174  l>u.  1  ph.  Ij,  qt. 

2.  How  much  will  25  doz.  pocket-knives  cost,  at  £3  2  s.  6  d.  a 
dozen?  &78  2s.6d. 

(3)                                       (4)                            (5) 
22  cd.  7  cd.  ft.  12  cu.  ft.              2°  43'    19"              5  mi.  37  ch.  56  1. 
9 m_ 14^ 

6.  What  is  the  weight  of  6  sets  of  silver  forks,  each  fork  weigh- 
ing 1  oz.  15  pwt.  12  gr.  ?  5  Tb.  3  oz.  18  pwt. 

7.  A  stone-mason  contracts  to  build  the  cellar  walls  for  13  dwell- 
ings. If  it  takes  7  cd.  5  cd.  ft.  4  cu.  ft.  for  each  cellar,  how  much 
stone  will  it  take  for  all  of  them  ?  99  cd.  4  cd.  ft.  ^  cu.  ft. 

8.  If  in  digging  the  cellars, 76  cu.  yd.  15  cu.  ft.  of  earth  be  taken 
from  each,  how  much  earth  will  be  taken  from  all  of  them  ? 

995  cu.  yd.  6  cu.  ft. 

9.  A  butcher  slaughtered  18  sheep,  and  their  average  weight 
was  35  lb.  15  oz.     What  was  their  total  weight  ? 

10.  A  train  of  63  coal  cars  was  loaded  at  a  coal  mine  in  Penn- 
sylvania, 3  T.  5  cwt.  2  qr.  of  coal  being  put  upon  each  car.  How 
much  coal  did  the  train  carry  ? 

11.  Multiply  27  mi.  218  rd.  4  yd.  2  ft.  8  in.  by  145. 

4,014  mi.  58  rd.  4  yd.  2  ft.  8  in. 

12.  If  5  men  can  make  38  rd.  5  yd.  of  post-and-rail-fence  in  a 
day,  how  much  fence  can  they  build  in  30  days  ? 

3  mi.  207  rd.  1  yd.  1ft.  6  in. 

13.  A  piece  of  land  near  a  city  was  divided  into  38  lots,  each 
containing  50  sq.  rd.  24  sq.  yd.  Horw  much  land  was  there  in  tho 
piece  ?  12  A.  10  sq.  rd.  ^.5  sq.  yd. 


156  COMPOUND    NUMBERS. 

14.  If  a  steam-boat  runs  a  mile  iii  4  min.  30  sec,  in  how  long  a 
time  will  it  make  a  trip  of  295  miles  ?  221i.l  min.  SO  sec. 

15.  If  a  housekeeper  uses  on  an  average  1  gal.  3  qt.  3  gi.  of 
molasses  in  a  month,  how  much  will  she  use  in  a  year  ?   21  gal.  S  qt. 

16.  A  farmer  drew  45  loads  of  hay  to  market,  and  each  load 
weighed  1  T.  375  lb.     How  much  hay  did  he  draw  ? 

17.  If  school  13  in  session  5  h.  25  min.  each  day,  how  long  is  it 
in  session  during  a  term  of  17  weeks  of  5  school  days  each  ? 

18.  If  the  rate  of  speed  of  a  railroad  train  is  25  mi.  315  rd.  an 
hour,  how  far  will  it  run  in  24  hours  ? 

19.  How  much  wine  in  eight  casks,  if  each  contains  28  gal. 
2  qt.  1.5  pt.  ? 

20.  Charles  is  7  yr.  251  da.  old,  and  his  grandfather  is  9  times  as 
old  as  he.     How  old  is  his  grandfather  ? 


SECTION  VI. 

CJ^SE     I. 
258*    The  Divisor  au  Abstract  Number. 
Ex.  Divide  46  mi.  126  rd.  1.5  yd.  by  9. 

Explanation. — ^We  write  solution. 

the  dividend  and  divisor  as      -/^g  mz.  126  rd.  1.5  yd.  [9 
in  integers,  and  commenc-        5  mi.  ^P  rd.    3  yd.  Oft.  8  in. 
ing  at  the  left,  we  divide 

tlie  units  of  each  denomination  of  the  dividend  by  the 
divisor,  in  order,  from  the  highest  to  the  lowest.  The 
dividend  being  a  concrete  number,  the  quotient  must  be  a 
concrete  number  (see  109,  IV.)  ;  the  quotient  arising  from 
dividing  the  units  of  any  denomination  must  be  of  the 
same  denomination  (see  109,  V.)  ;  and  any  partial  remain- 
der must  be  of  the  same  denomination — or  order  of  units — 
as  the  partial  dividend  used  (see  109,  VIII.).     One  ninth 


DIVISION.  157 

of  46  mi.  is  5  mi.  witli  a  remainder  of  1  mi.  We  write  the 
5  mi.  as  the  miles  of  the  quotient,  reduce  the  1  mi.  remain- 
der to  rods,  and  to  it  add  the  126  rd.,  making  446  rd. 
One  ninth  of  446  rd.  is  49  rd.  with  a  remainder  of  5  rd. 
We  write  the  49  rd.  as  the  rods  of  the  quotient,  reduce  the 
5  rd.  remainder  to  yards,  and  to  it  add  the  1.5  yd.,  making 
29  yd.  One  ninth  of  29  yd.  is  3  yd.  with  a  remainder  of 
2  yd.  We  write  the  3  yd.  as  the  yards  of  the  quotient,  and 
reducing  the  2  yd.  remainder  to  feet,  we  have  6  ft.  One 
ninth  of  6  ft.  is  no  whole  feet ;  we  therefore  write  0  ft.  in 
the  quotient,  and  reducing  the  6  feet  to  inches,  we  have  72 
in.  One  ninth  of  72  in.  is  8  in,,  which  we  write  as  the 
inches  of  the  quotient.  The  result,  5  mi.  49  rd.  3  yd.  8  in., 
is  the  quotient  required. 

In  integers  and  decimals,  the  units  of  each  order  are 
divided  separately,  and  the  units  in  any  partial  remainder 
are  caUed  10  times  as  many  units  of  the  next  lower  order. 

In  compound  numbers  the  units  of  each  denomination 
are  divided  separately,  and  the  units  in  any  partial  remain- 
der are  changed  to  units  of  the  next  lower  denomination 
by  reduction.    That  is, 

I.  In  division  of  integers  and  decimals,  any  partial  remain- 
der is  tens  of  the  order  of  units  next  lower  than  the  partial 
dividend  used  ;  And, 

II.  In  division  of  compound  numbers,  each  unit  of  any 
partial  remainder  is  as  many  times  1  of  the  next  lower  denom- 
ination, as  there  are  ones  of  the  lower  denomination  in  a  unit  or 
1  of  the  partial  dividend  used. 

PMOBLEMS. 

1.  A  silver-ware  manufacturer  used  5  lb.  6  oz.  12  pwt.  of  silver 
in  making  9  goblets.     How  much  silver  did  he  use  for  each  ? 

2.  In  settling  an  estate,  a  farm  of  184  A.  46.25  sq.  rd.  was  divided 
equally  among  15  heirs.   How  much  land  did  each  heir  receive  ? 

12  A.  4S.75  sq.  rd. 


158  COMPOUND    NUMBERS. 

3.  A  farmer  cut  11  T.  17  cwt.  of  hay  from  6  acres  of  meadow. 
What  was  the  yield  per  acre  ?  1  T.  1,950  IK 

(4)  (5) 

56  gal.  2  qt.  1  pt.  1  gi.  [  7  46  mi.  230  rd.  4.5  yd.  [  9 

6.  If  2,864  cu.  yd.  24  cu.  ft.  of  stone  are  used  in  making  586  rd. 
of  Macadamized  road,  how  much  stone  will  be  used  in  making 
1  rd.  ?  Jt  cu.  yd.  21i,  cu.  ft. 

C^SE    II. 
259.    The  Divisor  a  Concrete  Number. 

$4  are  contained  in  $12,  3  times  ;  $.04  in  $.12,  3  times  ; 
4  lb.  in  12  lb.,  3  times  ;  4  oz.  in  12  oz.,  3  times  ;  et-c. 
That  is. 

We  can  divide  dollars  by  dollars,  cents  by  cents,  pounds 
by  pounds,  ounces  by  ounces,  etc.  But  we  can  not  divide 
dollars  by  cents,  nor  pounds  by  ounces.  For  $12  -^  $.04  = 
neither  $3  nor  $.03  ;  so,  also,  12  lb.  -f-  4  oz.=  neither  3  lb. 
nor  3  oz.     Hence, 

Only  concrete  numbers  of  the  same  denomination  can  be 
divided,  the  one  by  the  other. 

Ex.  1.  How  many  times  are  4  lb.  9  oz.  contained  in 
27  lb.  6  oz.  ? 

Explanation. — Since     both    divi-  solution. 

dend    and   divisor    are   compound      ^^  ^^«  ^  oz.  =  4^8  oz. 
numbers,  we  reduce  them  to  simple         ^  ^^'  ^  ^^'  —    '^^  ^^' 

73  oz, 
nomination,  (ounces),  and  divide  as      j^S8 

in  integers. 

Ex.  2.  Divide  86  yd.  1  ft.  by  8  ft.  4  in. 

Explanation. — Since  the  lowest       „^     ,    ^^^l'^^^^^  ^^^  . 
,  .,...,.,  .        86  yd.  1ft.  =3.108  m. 

denomination  m  either  term  is        g  %     t  ^^^  _    ^  joo  in 

inches,  we  first  reduce  both  terms 

to  inches,  and  then  divide  as  in  3,108  in.  |  100  in, 

integers  and  decimals.  31.08 


DIVISION.  -   159 

Ex.  3.  105  wk.  are  how  many  solution. 

times  3  wk.  4  da.  ?  105  lok.  =  735  da. 

Explanation.  — After  dividing       ^  ^^^-  -^  ^^-  —    ^^  ^^• 
all  the  units  of  the  dividend,  we 
annex  a  decimal   cipher  to  the 
remainder,     and    continue     the 
division  as  in  decimals. 


735  da.  I  25  da, 
50  -ZTT" 


235 
225 


PROBLEMS,  100 

7.  I  have  a  measure  that  is  5  ft.  8  -^^^ 
in.  long.    How  many  times  the  length 

of  my  measure  is  a  pole  10  yd.  7  ft.  4  in.  long  ? 

8.  If  a  man  feeds  his  horse  1  pk.  6  qt.  of  oats  a  day,  how  long 
will  5  bu.  1  pk.  last  him  ?  12  days. 

9.  A  joiner  used  2  gro.  7  doz.  3  screws  in  hanging  and  trim- 
ming the  doors  of  a  house,  using  1  doz.  5  screws  to  each  door. 
How  many  doors  were  in  the  house  ?  22. 

10.  At  how  many  loads  can  a  teamster  draw  93  cd.  1  cd.  ft.  8 
cu.  ft.  of  wood,  drawing  1  cd.  2  cd.  ft.  8  cu.  ft.  at  a  load  ? 

11.  How  many  demijohns,  each  containing  3  gal.  2  qt.  1  pt,,  can 
be  filled  from  97  gal.  3  qt.  1  pt.  of  wine  ?  27. 

260.  Upon  the  principles  deduced  in  258,  259,  is  based 
the 

mule  for  division  of  Compound  JV^umders. 

I.  When  the  divisor  is  an  abstract  number. 

1.  Divide  the  units  of  each  denomination  separately,  and 
ivrite  the  several  results  for  the  same  denominations  of  the 
quotient. 

2.  Reduce  each  partial  remainder  to  the  next  lower  denomi- 
nation, and  add  to  it  the  units  of  that  denomination,  for  the  next 
partial  dividend. 

11.  When  the  divisor  is  a  compound  or  a  concrete  number. 
Reduce  both  dividend  and  divisor  to  the  lowest  denomination 
contained  in  either,  and  divide  as  in  integers  and  decimals. 


160  COMPOUND    NUMBERS. 

PJt  OBIjEMS. 

13.  Divide  71  mi.  237  rd.  3  yd.  1  ft.  6  in.  by  9. 

7  mi.  310  rd.  ^  yd.  2  ft, 

13.  A  wood-chopper  cut  68  cd.  3  cd.  ft.  of  wood  in  26  days. 
How  much  did  he  cut  per  day  ?  2  cd.  3.5  cd.  ft. 

14.  An  importer  paid  £663  15  s.  for  50  gold  watches.  What  did 
they  cost  apiece  ?  £13  5  s.  6  d.     . 

15.  A  farmer  raised  488  bu.  1  pk,  of  barley  from  14  acres  of 
land.    What  was  the  yield  per  acre  ? 

34.875  lu.j  or  3^  hi.  3  pk.  4  qt. 

16.  If  a  vessel  sails  250  mi.  in  2  da.  4  li.  5  min.,  what  is  the 
average  rate  of  speed  ?  1  mi.  in  12  min.  30  sec. 

17.  If  a  paver  in  24  days  can  put  down  76  sq.  rd.  15  sq.  yd.  of 
pavement,  how  much  can  he  put  down  in  one  day  ? 

3  sq.  rd.  5  sq.  yd.  6  sq.  ft. 

18.  If  a  train  of  27  cars  carry  57  T.  1  cwt.  2  qr.  24  lb.  of  ii'on 
ore,  what  is  the  average  per  car  ? 

19.  A  family  consumes  7  lb.  11  oz.  of  meat  each  week;  how  long 
will  it  take  them  to  consume  192  lb.  3  oz.  ?  25  weeks. 

20.  In  256,728  cu.  in.  how  many  gallons  liquid  measure  ?  How 
many  quarts  dry  measure  ? 

21.  If  6  men  in  12  days  mow  86  A.  64  sq.  rd.  of  grass,  how 
much  will  1  man  mow  in  1  day  ?  1  A.  32  sq.  rd. 

22.  If  a  railroad  train  runs  144  mi.  291  rd.  in  5.5  h.,  how  far 
does  it  run  per  hour  ?  26  mi.  Ill  rd.  1ft.  6  in. 

23.  How  many  silver  spoons,  each  weighing  1  oz.  9  pwt.,  can  be 
made  from  12  lb.  18  pwt.  of  silver  ? 

24.  How  long  must  a  field  be  to  contain  14  A.,  if  it  is  35  rd. 
wide? 

25.  How  many  bars  of  railroad  iron,  each  18  ft.  long,  will  be 
required  for  a  railroad  521  miles  long  ? 

26.  How  many  rolls  of  wall-paper,  20  in.  wide  and  9  yd.  long, 
will  be  required  to  paper  the  walls  of  a  room  14  by  16  ft.  and  9  ft. 
high,  no  allowance  being  made  for  openings  in  the  walls  ?       12. 


REVIEW    PROBLEMS.  161 

SECTION  VII. 

I.  How  many  yards  of  carpeting  will  be  required  to  cover  a 
floor  24  ft.  long  and  18  ft.  wide  ?  J^S. 

3.  A  note  dated  June  17,  1865,  was  made  payable^  January  10, 
1866.    How  long  had  it  to  run  ?  f '^'        £%  * 

3.  How  many  bushels  of  wheat  can  a  farmer  store  in  a  molasses 
hogshead  which  will  hold  138  gallons  ?  (See  248,  Note  3.)  IS  lu.  SpTc, 

4.  A  manufacturer  of  patent  medicine  puts  1  3  23  10  gr.  of  cal- 
omel into  each  bottle  of  medicine.  How  much  calomel  does  he  use 
for  50  dozen  bottles ?  llrb.51  4z. 

5.  If  I  start  at  latitude  15°  35'  40"  north,  and  travel  due  north 
2,159  geographic  miles,  in  what  latitude  will  I  be  ? 

61°  34'  40"  north. 

6.  What  is  the  capacity  in  bushels  of  a  bin  8  ft.  long,  7  ft.  wide, 
and  0  ft.  deep  ?  270. 

7.  A  gardener  raised  31  bu.  1  pk.  5  qt.  of  marrowfat  peas  for 
seed,  and  put  them  up  in  papers,  each  holding  .25  pt.  How  many 
pa,pers  did  he  put  up  ? 

0.  How  many  pieces  of  ribbon,  each  1  eighth  of  a  yard  long,  can 
be  cut  from  5  pieces,  each  containing  35  yd.  3  qr.  ? 

9.  An  apothecary  pays  $3.50  per  pound  avoirdupois  for  6  pounds 
of  rhubarb.  If  he  sells  it  in  prescriptions,  at  the  rate  of  $.30  an 
ounce,  how  much  does  he  gain  ? 

10.  What  is  the  weight  of  1,250  barrels  of  flour? 

1S2.5  T.  =  m  T.  10  cwt. 

II.  How  many  pump  logs,  each  12  ft.  long,  will  it  require  to 
bring  water  to  my  house  from  a  spring  1.375  miles  distant  ? 

13.  Plow  many  bunches  of  lath  will  be  required  for  the  walls  of  a 
room  18  ft.  long,  15  ft.  wide,  and  13  ft.  liigh  ? 

13.  In  digging  a  ditch  130  rd.  long  and  3  ft.  wide,  1,330  cu.  yd. 
of  earth  were  removed.     How  deep  is  the  ditch  ?  6  ft. 

14.  How  many  more  minutes  in  the  summer  months  than  in  the 
winter  months  of  a  common  year  ? 


162  COMPOUND     NUMBERS. 

15.  How  many  bunches,  each  containing  500  shingles,  -will  be 
required  to  cover  a  roof,  each  side  of  which  is  75  ft.  long  and  23.5 
ft.  wide  ?  J^O.5. 

16.  How  many  bricks  lying  flatwise  will  be  required  for  a  walk 
25  rd.  4  ft.  long  and  5  ft.  wide  ?  9371.25. 

17.  What  is  the  difference  between  a  figure  which  contains  .5  of 
a  sq.  ft.  and  one  which  is  .5  of  a  foot  square  ? 

18.  How  many  times  will  a  wagon  wheel  12  ft.  6  in.  in  circum- 
ference turn  round  in  going  11  mi.  28  rd.  ? 

19.  How  many  more  farthings  in  £19  4.5  d.  than  in  £17  19  s. 
8.875  fai-.  ? 

20.  A  merchant  bought  15  pieces  of  merino,  each  piece  contain- 
ing 42  yd.,  at  2  s.  per  yd.     What  was  the  amount  of  his  bill  ? 

21.  A  stationer  paid  7  s.  a  gross  for  1,080  pen-holders.  How 
much  did  they  cost  him  ?  &2  12  s.  Gd. 

22.  I  own  a  tract  of  Western  land  215  rd.  long  and  140  rd.  wide. 
How  many  acres  in  the  tract  ?  188.125. 

23.  When  hay  sells  at  $1.37|-  per  hundred,  what  is  the  price  per 
ton? 

24.  How  many  cords  of  stone  will  be  required  to  lay  a  wall  218 
ft.  long,  3  ft.  high,  and  1.5  ft.  thick  ?  7  cd.  5  cd.ft.  5  cu.ft. 

25.  A  pile  of  wood  183  x  6  x  4  ft.  contains  how  many  cords? 

26.  .02875  bu.  is  what  decimal  of  a  quart  ?  .92  gt. 

27.  How  many  bricks  will  be  required  for  the  two  side  walls  of 
a  building  50  ft.  long,  20  ft.  high,  and  1  ft.  thick  ? 

28.  What  is  the  capacity  of  a  cistern  14  ft.  long,  11  ft.  wide,  and 
7.5  ft.  deep  ?  137  hU.  9  gal. 

29.  A  pile  of  wood  which  contains  12.25  cd.,  is  56  ft.  long  and  8 
ft.  high.      What  is  its  width  ? 

30.  A  man  owns  3.5  A.  of  land.  If  he  lays  it  out  into  village 
lots,  each  5  by  8  rd.,  how  many  lots  will  he  have  ? 

31.  What  is  the  entire  weight  of  15  bar.  of  pork,  12  bar.  of  flour, 
6  casks  of  nails,  and  8  bar.  of  K  Y.  salt  ?  ^  T.  192  11. 

32.  How  many  quart  cupfuls,  tin  measure,  in  2  bu.  3  pk.  of 
chestnuts.     (See  249,  Note  3.) 

33.  How  much  will  18  barrels  of  pork  cost,  at  $.12|^  per  pound? 


REVIEW    PROBLEMS.  163 

34.  A  milkman  furnishes  420  qt.  of  milk  daily  to  tlie  customers 
on  his  route.  How  many  barrels  of  miilk  does  he  furnish  in  2 
weeks  ? 

35.  A  man  bargained  for  3  qt.  of  blackberries  daily  during  the 
blackberry  season,  which  lasted  21  days.  How  many  berries  did 
he  buy  ? 

36.  How  many  square  yards  in  the  four  walls  of  a  room  which  is 
40  ft.  long,  30  ft.  wide,  and  20  ft.  high  ? 

37.  How  many  square  yards  in  the  ceiling  of  the  same  room  ? 

38.  A  coal  dealer  bought  225  tons  of  coal,  at  $4.75  per  gross  ton, 
and  retailed  it  for  $6.50  per  net  ton.     What  was  his  gain  ?  $569.25, 

39.  How  much  land  is  there  in  a  field  that  is  11  x  13  ch.  ? 

40.  Reduce  1,000,000  cu.  in.  to  higher  denominations. 

41.  Reduce  1,000,000  sq.  in.  to  higher  denominations. 

25  sq.  rd.  14.75  sq.  yd.  5  sq.  ft.  64  sq.  in., 
or  25     "      15  "      3    "      28    " 

42.  A  farmer  raised  5  acres  of  potatoes,  which  yielded  175  bush- 
els to  the  acre.  He  sold  them  to  a  grocer  for  $2.75  a  barrel.  How 
much  did  he  receive  for  them  ? 

43.  How  many  days  were  there  in  the  last  century  ? 

44.  How  many  seconds  in  the  circumference  of  a  cart  wheel  ? 

45.  A  printer  used  3  rm.  2  quires  12  sheets  of  paper  for  quarter- 
sheet  posters.     How  many  posters  did  he  print  ?  6,000. 

46.  10,000  silver  dollars,  of  412.5  gr.  each,  weigh  how  many 
pounds  ?  716  lb.  1  oz.  15  pwt. 

47.  A  farmer  sold  25  T.  625  lb.  of  hay  for  $8.37|-  per  load  of 
1,125  lb.     How  much  did  he  receive  ? 

48.  If  you  were  to  count  80  1-dollar  bills  every  minute,  for  10 
hours  a  day,  how  long  would  you  be  in  counting  $1,000,000  ? 

49.  A  fruit-dealer  bought  3  bu.  3  pk.  (=3.75  bu.)  of  cranberries, 
at  $6.75  per  bushel,  and  retailed  them  at  $.25  per  quart,  tin  meas- 
ure.    What  was  his  profit  ?  $9.68%* 

50.  A  produce  dealer  buys  5,600  bu.  of  oats  in  St.  Louis,  Mo.,  <^ 
$.37|^,  and  sells  them  in  New  Haven,  Conn.,  ®  $.75.  How  much 
are  his  profits,  freights  being  $.31|-  per  bu.,  payable  at  St.  Louis  ? 

$l,JiO0. 


CHAPTER  4, 
FACTORS  AND  MULTIPLES 


SECTION  I. 

^BF'IJ^I  TIOJV'S, 

261.  An  SJxact  divisor  of  a  number  is  any  factor  of 
that  number.     Thus,  2,  3,  4,  and  6  are  exact  divisors  of  12. 

262.  An  ^ven  JViimber  is  one  that  is  exactly  divis- 
ible by  2  ;  as  2,  4,  20,  36,  758. 

263.  An  Odd  JVumber  is  one  that  is  not  exactly  divis- 
ible by  2  ;  as  1,  3,  7,  29,  245. 

264.  A  Composite  JVzwiber  is  one  that  can  be  sepa- 
rated into  factors.  Thus,  18  is  a  composite  integer,  and  its 
factors  are  2  and  9,  or  3  and  6,  or  2,  3,  and  3. 

265.  A  !Przme  JVtmiber  is  one  that  can  not  be  sepa- 
rated into  integral  factors  ;  as,  3,  5,  7,  29,  257. 

Note. — ^When  the  factors  of  a  number  are  prime  numbers,  tliey  are 
called  Prime  Factors.  Thus,  4  and  6,  or  3  and  8,  or  2  and  12,  are  factors 
of  24;  but  the  prime  factors  of  24  are  2,  2,  2,  and  3. 

266.  A  Common  !Divisor  of  two  or  more  numbers  is 
any  factor  found  in  each  of  them.  Thus,  4  is  a  common 
divisor  of  24,  36,  and  48. 

NoTB. — Two  or  more  numbers  are  prime  to  eacli  other  when  they  have  no 
common  factor.    The  number  1  is  not  regarded  as  a  factor. 

267o  The  Greatest  Common  divisor  of  two  or  more 
numbers  is  the  greatest  factor  found  in  each  of  them. 
Thus,  12  is  the  greatest  common  divisor  of  24,  36,  and  48. 

268.  A  Mtdtipte  or  SJxact  !Dividend  is  a  num- 
ber of  which  a  given  number  is  a  factor.  Thus,  27  is  a 
multiple  of  9. 


CHANGES    OF    DIVIDEND    AND    DIVISOR.      165 

269.  A  Common  J}fultipte  of  two  or  more  numbers 
is  a  number  of  wliich  each  of  the  given  numbers  is  a  factor. 
Thus,  32  is  a  common  multiple  of  4,  8,  and  16* 

270.  The  Zeast  Comm07i  MulHpte  of  two  or  more 
numbers  is  the  least  number  of  which  each  of  the  given 
numbers  is  a  factor.  Thus,  30  is  the  least  common  multiple 
of  3,  5,  10,  and  15. 


SECTION  n. 

271.  The  value  of  the  quotient  depends  upon  the  values 
of  both  dividend  and  divisor*  Hence,  any  change  in  either 
of  these  terms  must  produce  a  change  in  the  quotient. 

C.A.SE    I. 


Changes  of  Dividend. 


272.  The  quotient 
of  30  -^  5  is  6.  n 
we  multiply  the  div- 
idend by  2,  and  di- 


S0_  \5 
6 


2x30  [5 


60_  \5 
12=2x6 


7x30  {5 
210  \5 
J^=z7x6 


vide  the  product  (60)  by  5,  the  quotient  is  12,  or  2  times  6. 
Again,  if  we  multiply  the  dividend  by  7,  and  divide  as 
before,  the  quotient  is  42,  or  7  times  6.     Hence, 
Multiplying  the  dimdend  multiplies  the  quotient. 


3_0_  \3 
10 


30-^2  \ 


W  {3 
5=10- 


30^5  [  3 


2=10-^5 


273.  The  quo- 
tient of  30  -^  3  is 
10  ♦  If  we  divide 
the  dividend  by 
2,  and  then  divide  the  quotient  (15)  by  3,  the  result  is  5,  or 
10  -f-  2*  Again,  if  we  divide  the  dividend  by  5,  and  then 
divide  the  quotient  (6)  by  3,  the  result  is  2,  or  10  -~  5. 
Hence, 

Dividing  the  dividend  divides  the  quotient. 


166 


FACTORS    AND    MULTIPLES. 


C^SE     II. 
Changes  of  Divisor. 


120  \  15 


120  [2 X  15 


120  [SO 
Jf=8-^2 


m 

120 


J^xl5 


2=8^4 


274.  The  quotient 
of  120-f-15  is  8.  If 
we  multiply  the  di- 
visor by  2,  and  then 
divide  120  by  the  product  (30),  the  quotient  is  4,  or  8  -^  2. 
Again,  if  we  multiply  the  divisor  by  4,  and  divide  as  before, 
the  quotient  is  2,  or  8  -h  4.     Hence, 


3Iultiplying  the  divisor  divides  the  quotient. 


90  lis 


90_  \18-^2 
10=2x5 


90  118~6 
90  [S 
30=6x5 


275.  The  quotient  of 
90-^18  is  5.  If  we 
divide  the  divisor  by  2, 
and  then  divide  90  by 
the  quotient  (9),  the  result  is  10,  or  2  times  5,  Again,  if 
we  divide  the  divisor  by  6,  and  then  divide  the  dividend 
(90)  as  before,  the  result  is  30,  or  6  times  5.     Hence, 

Dividing  the  divisor  multiplies  the  quotient. 


C^SE    III. 
Like  Changes  of  Dividend  and  Divisor. 


S2  {8 
1 


2x32  [2x8 
64.  [16 


5x32  y5x8 
160  \^JfO 


276.  The  quotient 
of  32  -h  8  is  4.  If 
we  multiply  both 
dividend  and  divi- 
sor by  2,  and  divide  the  new  dividend  (64)  by  the  new 
divisor  (16)  the  quotient  is  4,  the  same  as  before.  Again, 
if  we  multiply  both  terms  by  5,  and  then  divide  as  before, 
the  quotient  is  still  4.     Hence, 

Multiplying  both  dividend  and  divisor  by  the  same  number 
does  not  change  the  quotient. 

277.  It  has  already  been  shown,  in  Art  107,  that 
Dividing  both  dividend  and  divisor  by  the  same  number  does 

not  change  the  quotient. 


CHANGES    OF    DIVIDEND    AND    DIVISOR.      167 

278.  These  three  cases  establish  the 

General  ^ri7iciples  of  Divisi07i, 

I.  Tae  quotient  is  multiplied  by  multiplying  the  dividend  or 
dividing  the  divisor. 

n.  The  quotient  is  divided  by  dividing  the  dividend  or  mul- 
tiplying the  divisor. 

III.  The  quotient  is  not  changed  by  either  multiplying  or 
dividing  both  dividend  and  divisor  by  the  same  number. 

Sec  Manual. 

Cancellation. 

279.  Ex.  What  is  the  quotient  of  4  x  75  divided  by  4  x  3  ? 
Explanation.  —  When   division  is  exact,  solution. 

the  quotient  consists  of  that  factor  of  the      U^'^^  l-^^'^ 
dividend  not  common  to  both  dividend  and  15  \S 

divisor.    In  this  example  4  is  a  factor  of  both  ^ 

dividend  and  divisor.  And,  since  the  quo- 
tient is  not  changed  by  dividing  both  dividend  and  divisor 
by  the  same  number  (278,  m.),  we  divide  4  x  75  and 
4  X  3  by  4, — or,  which  is  the  same  thing,  we  omit  the  factor 
4  from  both  terms, — and  divide  75,  the  remaining  factor  of 
the  dividend,  by  3,  the  remaining  factor  of  the  divisor. 

Cancellation  is  the  process  of  omitting  or  striking 
out  equal  factors  from  the  dividend  and  divisor. 

280.  From  the  solution  and  explanation  of  the  last  ex- 
ample, we  see  that 

A  factor  is  cancelled  by  dividing  both  dividend  and  divisor 
by  that  factor. 

Ex.  1.  Divide  210  by  35.     •  solution. 

Explanation.— We  can-  ^^''^^'  ^  l?£   ^'''''''^ 

eel   the  common   factor,    5,         -^ew  Dividend,     4^  (^7       New  Divisor. 

by  dividing  both  dividend  q    Quotient 

and  divisor  by  5  ;  and  then 

divide  the  new  dividend,  42,  by  the  new  divisor,  7. 


168  FACTORS    AND    MULTIPLES. 

Ex.  2.  Divide  21  x  64  by  56. 

Explanation. — Canceling  tlie  common  fac-  ^/°^^"°r^ 

tor,  7,  we  have  3  x 64  for  a  new  dividend,  "    ^^  ^— 

and  8  for  a  new  divisor.    Then,  canceling  SxGJf.  \^8 

the  common  factor,  8,  we  have  3x8  for  a  8x8  {^1 

dividend  and  1  for  a  divisor.     The  product  £/ 
of  3  X  8,  or  24,  is  the  required  quotient. 

Note. — From  this  solution  and  explanation  wc  learn  that, 
When  either  dividend  or  divisor  is  canceled,  a  1  helongs  in  its  place. 

Ex.  3.  Divide  the  product  of  24,  80,  9,  and  12.8,  by  the 
product  of  3.2,  144,  and  16. 

Explanation. — The  fac-  ^'^-st  solution. 

tors  of  the   divisor  may  r     q          3                 \ 

be  written  at  the  right  ^^^  x  ^0  x  0  x  XU  [  U  x  XM  x  H 

of  those  of  the  dividend,  5x8x2  =  30 
as    shown  in  the   First 

Solution;  or  under  those  second  solution. 

of  the  dividend,  as  shown  r     q          a 

in  the  Second  Solution.  ^^  x  ^0  x  0  x  lt$  _  ^     o 

"When   a   factor  is   can-  ~~r~      ~r:     rr — —ox8x2=80 

celed,    we  draw  an   ob-  ^ 

lique  line  across  it.  ^ 

:PJtOBJLEMS, 

1.  The  dividend  is  714,  and  the  divisor  42.  What  is  the  quotient  ? 

2.  What  is  the  quotient  of  21  x  13  divided  by  7  ? 

3.  How  many  times  is  11  x  15  contained  in  825  ? 

4.  Divide  28  x  7.2  by  16.  12.6. 

5.  The  factors  of  a  dividend  are  8,  25,  and  .45 ;  and  of  a  divisor, 
15,  2,  and  1.2.     What  is  the  quotient  ?  2.5. 

6.  How  many  tons  of  hay   at  $10  a  ton,  must  be  given  in  ex- 
change for  16  tons  of  coal  at  $5  a  ton  ? 

7.  In  how  many  days  can  45  men  do  as  much  work  as  63  men 
can  do  in  35  days?  -4^. 

8.  If  a  mechanic  can  earn  $88  in  28  days,  how  much  can  he  earn 
in  42  days  ? 


PROPERTIES    OF     COMPOSITE    NUMBERS.      169 

9.  I  rent  a  store  18  months  for  $2,400.  What  is  the  rent  per  year? 

10.  If  15  acres  of  land  produce  420  bushels  of  wheat,  how  much 
wheat  will  35  acres  produce,  at  the  same  rate  ? 

11.  A  ship's  crew  of  39  men  have  provisions  enough  to  last  them 
76  days.  If  the  crew  is  increased  to  57  men,  how  long  will  the 
j^rovisions  last  them  ?  ^3  days. 

12.  In  building  a  church,  9  bricklayers  laid  407,880  bricks  in  55 
days.  At  the  same  rate,  how  many  bricks  can  11  bricklayers  lay 
in  60  days  ? 

J  3.  If  a  telegram  of  2,790  words  can  be  transmitted  in  45  min- 
utes, how  many  words  can  be  telegraphed  in  72  minutes  ?    4j4^4- 


SECTION  in. 

1>1R0^jE^TIBS   OJF"  C0M^0SIT£J  jyvm:sb^s, 

281.  The  right-hand  figure  of  any  even  number  is  0,  2,  4, 
6,  or  8.     Hence, 

Peopekt^y  I.  Any  number  is  divisible  by  2^  when  its  right- 
hand  figure  is  0,  2,  Ji-,  Gy  or  8.     (See  2^2.) 

282.  Tlie  right-hand  figure  of  the  product  of  any  even 
number  of  times  5  is  0  ;  thus  2  x  5=10,  6  x  5  =  30,  14x5  = 
90.  The  right-hand  figure  of  any  odd  number  of  times  5  is 
5  ;  thus,  3  X  5=15,  7  x  5=35  ;  19  x  5  =  95.     Hence, 

Property  II.  Any  number  is  divisible  by  o,  ivkeii  its  right- 
hand  figure  is  0  or  5. 

283.  Any  number  exj)ressed  by  more  than  one  figin-e  may 
be  separated  into  two  parts,  one  of  which  is  a  multiple  of 
some  power  of  10,  and  the  other  is  ones.  Thus,  56  =  50  +  6, 
or  5  times  10'  and  6  ones  ;  256  =  200  +  56,  or  2  times  10- 
and  56  ones  ;  3256  =  3000  +  256,  or  3  times  10'  and  256 
ones  ;  and  so  on.     Hence, 

Property  III.  Any  number  is  divisible  by  any  power  of  2 
or  5,  when  the  number  expressed  by  as  many  of  its  right-hand 
figures  as  equal  the  index  of  the  power  is  divisible  by  2  or  5. 


170  FACTO  lis    AND    MULTIPLES. 

284.  The  number  54,678,  and  the  sum  of  the  numbers  5, 
4,  6,  7,  and  8  (the  digits  of  54,678),  are  divisible  by  3.  The 
number  12,348,  and  the  sum  of  the  numbers  1,  2,  3,  4,  and 
8,  are  divisible  by  9.     Hence, 

Property  IV.  Any  number  is  divisible  by  S  or  9,  luhen  the 
sum  of  its  digits  is  divisible  by  3  or  9. 

285.  We  can  divide  84  by  3,  and  the  result  (28)  by  7. 
We  can  also  divide  84  by  21,  the  product  of  3  times  7. 
Again,  we  can  divide  756  by  4,  the  result  (189)  by  3,  and 
this  result  (63)  by  7.  We  can  also  divide  756  by  all  the 
successive  divisors,  4,  3,  and  7,  and  by  their  product,  84. 
Hence, 

Property  V.  Any  number  which  is  divisible  by  two  or  more 
factors  successively,  is  also  divisible  by  each  of  the  factors,  and 
by  their  product. 

Note. — A  number  may  be  divisible  by  two  or  more  factors,  and  not  be 
divisible  by  their  product.  Thus,  24  is  divisible  by  8  and  by  12,  but  not 
by  their  product,  96. 

286.  If  we  divide  any  number  by  one  of  its  prime  fac- 
tors, and  divide  the  result  by  another  prime  factor,  and  so 
on,  until  the  quotient  is  1,  we  shall  use  all  the  prime  factors 
of  the  number  for  divisors.  Thus,  72  -^  2  =  36,  36  -=-  2  = 
18,  18  -^  2  =  9,  9  ^  3  ==  3,  3  -^  3  =  1.  The  factors  used  as 
divisors  are  2,  2,  2,  3,  3;  and  their  product,  72,  is  divisible 
by  the  product  of  any  number  of  these  factors.     Hence, 

Property  VI.  Any  number  is  divisible  by  the  product  of  any 

two  or  more  of  its  prime  factors. 

287.  The  factors  of  6  (2  and  3)  are  contained  in  12 
(=2  X  6),  18  (=  3  X  6),  30  (-  5  X  6),  54,  96,  or  any  num- 
ber of  times  6.     Hence, 

Property  VII.  Any  factor  of  a  composite  number  is  con- 
tained in  any  number  of  times  that  number. 


PRIME    NUMBERS.  171 

288.  Since  35  is  5  times  7,  and  21  is  3  times  7,  35  +  21, 
or  56,  is  5  times  7  +  3  times  7,  or  8  times  7  ;  and  35  —  21, 
or  14,  is  5  times  7  —  3  times  7,  or  2  times  7.     Hence, 

Property  VIII.  Any  factor  common  to  two  numbers  is  also 

a  factor  of  their  sum,  and  of  their  difference. 

Note. — These  properties  apply  more  generally  to  uumbers  in  the  deci- 
nml  scale ;  but  to  a  limited  extent  to  compound  numbers  also. 


SECTION    IV. 

289.  All  even  numbers  except  2,  and  all  odd  numbers  end- 
ing in  5  except  5,  are  composite.     (See  262,  282.)     Hence, 

The  right-hand  figure  of  a  prime  number  is  1,  3^  7^  or  9. 

290.  Since  2  is  a  factor  of  4  and  of  6,  it  is  a  factor  of  2  + 
6  and  of  4  +  6,  and  of  2  or  4  +  any  number  of  times  6 ; 
and,  since  3  is  a  factor  of  6,  it  is  a  factor  of  3  +  G,  and  of 
3  +  any  number  of  times  6.     (See  287,  288.) 

All  the  remainders  that  can  be  obtained  in  dividing  num- 
bers by  6  are  1,  2,  3,  4,  and  5  ;  and  we  have  just  shown 
that,  when  the  remainder  is  2,  3,  or  4,  the  number  divided 
is  composite.     Hence, 

When  any  prime  number  is  divided  by  6,  the  remainder  is 
1  or  5. 

PH  OBLEMS. 

1.  Which  of  the  numbers  19,  45,  67,  91  are  prime  numbers? 

2.  Of  the  numbers  103,  126, 131,  217,  which  are  prime  numbers  ? 

3.  Which  of  the  numbers  111,  133,  14.7,  149,  219,  3.42,  are  com- 
posite numbers  ? 

4.  Which  of  the  seven  numbers  293,  371,  385,  440,  524,  617,  and 
713  are  prime,  and  which  composite  numbers  ? 

5.  Determine  which  are  prime,  and  which  compos^^^Jf^'-tha 
numbers  911,  973, 103.3,  10.57,  3373,  3.407,  358.41^- r/'X:^^ 

6.  Find  all  the  prime  numbers  less  than  lOO./f^   j^ 


1= 


oar 


172  FACTORS    AND    MULTIPLES. 

SECTION  V. 
coMMOJV  :d iris O'^s. 

C^SE     I. 
Prime  Factors  or  Divisors. 

291.  Ex.  1.  \Vliat  are  the  prime  factors  of  1,260? 

Explanation. — Since  the  right-hand  figure  of       solution. 
1,260  is  0,  we  divide  by  the  prime  number  2       ^^^^  l^ 
(see  281);  and  for  the  same  reason,  we  divide        630  \2 
the  quotient  (630)  by  2.     Since  the  right-hand        g^^  i  ^ 
figure  of  the  second  quotient  (315)  is  5,  we  next  ~  . 

divide    by    the    prime    number    5    (see   282).  — 

Since  the  sum  of  the  digits  of  the  thii'd  quo-  ^\p 

tient   (63)    is    divisible    by   3,   we   divide   this  7 

quotient  by  3  (see  284) ;  and  for  the  same  reason, 
we  divide  the  fourth  quotient  (21)  by  3.     The  last  quotient 
(7)  is  a  prime  number.     The  product  of  the  divisors  2,  2, 
5,  3,  3,  and  the  last  quotient,  7,  is  1,260  ;  and  hence  they 
must  be  all  the  prime  factors  of  that  number.     (See  57.) 

Ex.  2.  What  are  all  the  factors  or  divisors  solittiox. 

of  30?  ^l^ 

Explanation. — ^We  first  find   all  the  prime  15  \^5 

factors  of  30  to  be  2,  5,  and  3,  and  each  of  ~~g 
these  is  a  divisor  of  30  (see  285).     Since  2,  3, 

and  5  are  prime  factors  of  30  ;  6,  or  2  times  3,  2x3=  6 

10,  or  2  times  5,  and  15,  or  3  times  5  are  also  2x5=10 

divisors  of  30  (see  286).     Hence,  all  the  fac-  <^x^=-^^ 
tors  or  divisors  of  30  are  2,  3,  5,  6,  10,  and  15. 

292,   Ultele  for  findi7ig  'Prime  I^actors, 

I.  Divide  the  number  by  any  prime  factor. 

II.  Divide  the  quotient  in  the  same  manner ;  and  so  on, 
till  a  quotient  is  obtained  that  is  a  prime  number.  The  divisors 
and  the  last  quotient  will  be  the  prime  factors  required. 


COMMON    DIVISORS.  173 

JPJt  OBZJEMS. 

1.  Find  tlie  prime  factors  of  540.  2,  2,  5,  3,  3,  S, 

2.  What  are  the  prime  factors  of  1,650  and  1,755  ? 

3.  Separate  1,836  into  its  prime  factors.  2,  2,  3,  3,  3,  17. 

4.  Separate  945  and  3,990  into  their  prime  factors. 

6.  What  are  all  the  factors  or  divisors  of  84  ?        Tliere  are  ten. 
6.  What  are  the  prime,  and  what  the  component  factors  of  164  ? 
There  are  3  prime^  and  2  component  factors. 

c^se:  II. 
Common  Factors  or  Divisors. 


293.  Ex.  1.  Find  a  common  divisor  of  soltttion. 

15,  25,  and  40.  Sx^ 

Explanation. — We  separate  the  given      10=5x2x2x2 
numbers  15,  25,  and  40,  into  their  prime 
factors,  and  find  that  5  is  a  divisor  of  each  number.  (See  268.) 

Ex.  2.  Find  all  the  common  divi-  souttion. 

sors  of  54  and  72.  H=2 xSxSxS 

Explanation.  —  We    separate    the  72=2x2x2x3x3 
numbers  54  and  72  into  their  prime  ^^^_^ 

factors,  as  in  Ex.  1,  and  we  find  that  gy^g—g 

2,  3,  and  3  are  common  factors,  and  2x3x8=18 
are  therefore  common  divisors  (see 

266).     Since  2,  3,  and  3  are  common  factors,  6  or  2  times 

3,  9  or  3  times  3,  and  18  or  2  times  3  times  3  are  also  com- 
mon factors  (see  286).  Hence,  2,  3,  3,  6,  9,  and  18  are  all 
the  common  divisors  of  54  and  72. 

FMOBLJEMS. 

7.  What  number  is  a  common  divisor  of  21  and  36  ?  3. 

8.  Find  a  common  divisor  of  4.5  and  10.5.  1.5. 

9.  What  are  the  prime  common  factors  of  20,  32,  50,  and  18  ? 

10.  Find  all  the  common  divisors  of  36,  42,  and  90. 

11.  How  many  common  divisors  have  64, 112,  48,  and  144  ?  Four. 


174  FACTORS    AND    MULTIPLES. 

C.A.SE    III. 
Greatest  Common  Factor  or  Divisor. 

FIRST     METHOD. 

294.  Ex.  Wliat  number  is  the  greatest  common  divisor 
of  252,  210,  and  168  ? 

Explanation. — We   separate  the  solution. 

nnmbers  252,  210,  and  168  into  ^52^2x2x3x3x7 
their  prime  factors,  as  in  Case  II.  %lzlllxlxlx7 
(see  293),   and  we  find,  that  2,  3,  2x3x7=Jf2 

and  7  are  the  only  prime  factors 

common  to  all  the  given  numbers.  Since  2,  3,  and  7  are 
all  the  common  prime  factors  of  the  given  numbers,  their 
product,  42,  is  the  greatest  common  factor  of  the  given 
numbers,  and  hence  is  the  greatest  common  divisor  re- 
quired.    (See  267.) 

SECOND     METHOD. 


i.  Ex.  What  is  the  greatest  common  divisor  of   21 
and  77? 

Explanation. — Since  the  common  di- 
visor of  two  numbers  can  not  be  greater 
than  the  less  number,  we  divide  77,  the 
greater  number,  by  21,  the  less,  and  ob- 
tain a  remainder  of  14.  If  14  is  a  divi- 
sor of  21,  it  is  also  a  divisor  of  77,  H 
which  equals  14  +  3  times  21  (see  287).  2 
Dividing  21,  the  first  divisor,  by  14,  the 
first  remainder,  we  obtain  a  remainder  of  7. 
a  divisor  of  14,  it  is  a  divisor  of  21,  which  equals  7  +  14, 
and  of  77,  which  equals  7  +  5  times  14  (see  287).  Divid- 
ing 14,  the  last  divisor,  by  7,  the  last  remainder,  we  find 
that  7  is  a  divisor  of  14.  Hence,  7  is  a  common  divisor  of 
21  and  77. 

Having  proved  that  7  is  a  common  divisor  of  21  and  77, 
we  must  now  prove  that  it  is  their  greatest  common  divisor. 


SOLUTION. 

77    21 

^l\7 

21 

u 

u 

'  1 

7 

>. 

NOTN 

T,  if  7  is 

COMMON    DIVISORS.  175 

Since  any  number  that  is  a  common  divisor  of  21  and  77, 
must  be  a  divisor  of  14  (see  288),  a  number  greater  than  14 
can  not  be  a  common  divisor  of  21  and  77.  Again,  any 
number  that  is  a  common  divisor  of  14  and  21,  must  be  a 
divisor  of  7  (see  288).  Hence,  7,  the  greatest  common 
divisor  of  itself  and  14,  is  the  greatest  common  divisor  of 
21  and  77. 

PROBLEMS. 

12.  What  is  the  greatest  common  divisor  of  56  and  84  ?       28. 

13.  What  is  the  greatest  common  divisor  of  .103  and  .153  ?  51. 

14.  Find  the  greatest  common  divisor  of  96,  130,  and  168. 

15.  What  is  the  length  of  the  longest  line  that  will  exactly  meas- 
ure two  fences,  one  96  rods  and  the  other  76  rods  long  ? 

296.  The  explanations  and  solutions  given  in  294,  295, 
are  sufficient  to  estabHsh  the  following 

^ule  for  fi7idi7iff  a  Greatest  Com?7ion  divisor, 

I.  Separate  the  numbers  into  their  prime  factors. 

II.  Multiply  together  all  the  factors  that  are  common.  Tlie 
product  will  he  the  greatest  common  divisor.     Or, 

Divide  the  greater  number  by  the  less,  the  first  divisor  by  the 
first  remainder,  the  second  divisor  by  the  second  remainder, 
and  so  on,  until  an  exact  divisor  is  obtained.  This  divisor 
will  be  the  greatest  common  divisor. 

Notes. — ^1.  By  the  Second  Method,  if  more  than  two  numbers  are  given, 
we  must  first  find  the  greatest  common  divisor  of  two  of  them,  then  of 
their  greatest  common  divisor  and  another  of  the  numbers,  and  so  on,  till 
all  the  given  numbers  have  been  used.  The  last  common  divisor  obtained 
will  be  the  greatest  common  divisor  of  all  the  given  numbers. 

2.  Only  abstract  numbers,  or  like  concrete  numbers  of  the  same  denom- 
ination, can  have  a  common  divisor. 

3.  The  common  divisor  of  two  or  more  concrete  numbers  may  be  either 
an  abstract  or  a  concrete  number.    (See  98,  I.,  II.,  III.) 


176  FACTORS    AND    MULTIPLES. 

r  It  OB  I,  EMS. 

16.  What  number  is  a  common  divisor  of  $25  and  $60  ?  5,  or  $5. 

17.  Find  a  common  divisor  of  16  A.  and  28  sq.  rd. 

18.  "What  is  the  greatest  common  divisor  of  135  and  225  ? 

19.  How  many  common  divisors  have  1  pk.  and  6  qt.  ?       One. 

20.  The  sides  of  my  garden  are  168  ft,  280  ft.,  182  ft.,  and  252 
ft.  What  is  the  greatest  length  of  boards  that  I  can  use  in  fencing 
it,  without  cutting  any  of  them  ?  IJ^feet. 

21.  If  283.5  yd.  Wamsutta,  567  yd.  N.  Y.  Mills,  and  445.5  yd. 
Lawrence  Mills  sheetings  are  in  whole  pieces  of  the  greatest  pos- 
sible equal  length,  how  many  yards  are  there  in  each  piece  ?     J^OM. 


SECTION  VI. 

COMMOjT  M77ZTITZBS. 

C^SES     I. 
Common  Multiples  or  Dividends. 

291.  Ex.  What  number  is  a  common  multiple  or  divi- 
dend of  15  and  24? 

Explanation. — Since  a  common  multiple  solution. 

of  15  and  24  is  a  number  of  which  both  15       ^^  x  2J^=360 
and  24  are  factors  (see  269),  and  since  any 
product  must  be  a  multiple  of  any  set  of  factors  which  will 
produce  it  (see  57,  285),  we  multiply  15  and  24  together. 
The  product,  360,  is  the  common  multiple  required. 

VU  OBIjEMS. 

1.  Find  a  common  multiple  of  3,  4,  and  6. 

2.  Find  a  common  multi^Dle  of  5,  7,  32,  and  10. 

3.  What  number  is  a  common  multiple  of  4.8,  9,  and  5.25  ? 

226.8 f  or  any  integral  nuniber  of  times  226.8. 

4.  What  number  is  a  common  multiple  of  $15,  $2,  and  $8.50  ? 

5.  Find  a  common  multiple  of  1  bu.  3  pk.,  1  jok.  4  qt.,  and  5  qt. 
1  pt.  (=  5.5  qt.)  115  hu.  2  pic. 


COMMON    MULTIPLES.  177 

CASE    II. 
Least  Common  Multiples  or  Dividends. 

298.  24  is  a  common  multiple  of  4  and  6,  because  all  tlie 
factors  of  4  (2  and  2)  and  of  6  (2  and  3)  are  also  primo 
factors  of  24  (2,  2,  2  and  3).  But  2,  2,  and  3  are  all  tlio 
prime  factors  required  to  produce  both  4  and  6  ;  and  a 
number  that  contains  only  the  prime  factors  2,  2,  3,  will 
also  contain  4  and  6  (see  285).  Multiplying  these  three  fac- 
tors together,  we  have  2  x  2  x  3  =  12  ;  and  since  12  con- 
tains all  the  prime  factors  of  4  and  6,  and  no  other  factors, 
it  is  their  least  common  multiple. 

Ex.  Find  the  least  common  multiple  of  18,  24,  and  30. 

Explanation. — Since  the  least 
common  multiple  of  18,  24,  and  solution. 

30  must  contain  only  the  prime  2/^=3x3x2x3 

factors   of    these  numbers   (see  30=2x3x5 

270),   we   separate   each  of  the 

numbers   into  its   prime   factors.  PHme  Factors  required. 

Since  24  has  the  greatest  num-  2x2x2x3x3x5=360 
ber  of  prime  factors,  we  next,  for 

convenience,  write  all  the  factors  of  24  (2x2x2x3)  in  a 
line.  Then,  comparing  the  factors  of  18  v*^ith  these  factors, 
we  find  that  we  have  all  the  factors  but  a  3  ;  and  we  write  a 
3  with  the  prime  factors  required.  Again,  comparing  the 
factors  of  30  with  the  prime  factors  required,  we  find  that 
we  have  all  the  factors  but  a  5  ;  and  we  write  a  5  with  the 
prime  factors  required.  We  now  have  all  the  prime  factors 
of  the  given  numbers,  and  no  others  ;  and  multii)lying  them 
together,  we  obtain  360,  their  least  common  multiple. 

PItOBJLEMS. 

6.  What  is  the  least  common  multiple  of  8,  12,  and  14  ?      168. 

7.  "What  is  the  least  common  multiple  of  $16  and  $20  ? 

8.  Find  the  least  common  multiple  of  .6  and  .8.  .2Ji.. 

9.  What  is  the  least  number  of  which  75,  225,  and  500  are  factors? 


178  FACTORS    AND    MULTIPLES. 

299.  (Hides  for  Finduig  Multiples. 

I.  For  a  Common  Multiple. 

AMtiply  the  numbers  together.  The  product,  or  any  number 
of  times  the  product,  will  he  a  common  multiple. 

n.  For  the  Least  Common  Multiple. 

1.  Separate  the  numbers  into  their  prime  factors. 

2.  3Iultiply  together  all  the  prime  factors  of  that  number 

having  the  greatest  number  of  prime  factors,  and  those  prime 

factors  of  the  other  -  numbers  not  found  in  the  factors  of  the 

number  taken.     The  product  will  be  the  least  common  midtiple. 

Note. — Only  abstract  numbers,  or  like  concrete  numbers  of  the  same 
denomination,  can  have  a  common  multiple.      See  Manual. 

PH  OB  i:  JEMS. 

10.  Find  a  common  multiple  of  36,  18,  24,  and  12. 

11.  Find  the  least  common  multiple  of  the  numbers  given  in 
Problem  10. 

12.  Find  the  least  common  multiple  of  2,  8,  4,  6,  8,  9,  12,  16,  18, 
24,  36,  48,  72.  lU- 

13.  What  number  is  the  least  common  multiple  of  9,  60,  45,  72, 
15,  35,  18,  12  ? 

14.  Find  the  least  common  multiple  of  the  nine  digits.        2,520. 

15.  What  is  the  least  common  multiple  of  2  yd.  1  ft.,  and  2  ft. 
8  in.  ?  3  rd.  2  yd.  6  in. 

16.  What  is  the  smallest  sum  of  money  for  which  a  person  can 
purchase,  either  oxen  at  $85  each,  or  cows  at  $35  each  ?  $595. 

17.  A  can  hoe  a  row  of  com  in  a  certain  field  in  30  minutes, 
B  can  hoe  a  row  in  20  minutes,  and  C  in  35  minutes.  What  is  the 
least  number  of  rows  that  each  can  hoe,  in  order  that  all  may 
finish  together  ? 

18.  In  a  factory  are  three  wheels,  which  revolve  once  in  25,  30, 
and  50  seconds  respectively.  What  is  the  least  time  in  which  all 
of  them  will  make  an  exact  number  of  revolutions  ?      2  min.  30  sec. 


SECTION  I. 

300.  The  number  one  half  may  be  obtained  by  dividing  1 
into  2  equal  parts  ;  one  third,  by  dividing  1  into  3  equal 
parts  ;  one  fourth,  by  dividing  1  into  4  equal  parts  ;  one 
fifth,  by  dividing  1  into  5  equal  parts  ;  and  so  on. 

Again,  two  tliirds  may  be  obtained  by  dividing  2  into  3 
equal  parts,  or  by  dividing  1  into  three  equal  parts  and 
taking  two  of  those  parts. 

Two  fourths  may  be  obtained  by  dividing  2  into  4  equal 
parts  ;  and  three  fourths,  by  dividing  3  into  4  equal  parts. 

Two  fifths,  three  fifths,  and  four  fifths  may  be  obtained  by 
dividing  2,  3,  and  4,  respectively,  into  5  equal  parts. 

Halves,  thirds,  fourths,  and  fifths  are  written  thus  : 
1  half,        ^, 

1  third,      A,         3  thirds,      f, 
1  fourth,    :|,         2  fourths,    f,         3  fourths,    f, 
1  fifth,       I,        2  fifths,       f ,        3  fifths,       I,  •      4  fifths,   f . 

When  1  is  divided  into  6  equal  parts,  the  parts  are  sixths  / 
when  into  7  equal  parts,  they  are  sevenths;  when  into  8 
equal  parts,  eighths  ;  and  when  into  9  equal  parts,  ninths. 

Sixths,  sevenths,  eighths,  and  ninths  are  written  thus  : 

1  sixth,    ^,          2  sevenths,  f,          1  eighth,    ^,          2  ninths,  f, 

3  sixths,  f,          3  sevenths,  f,          3  eighths,  f,          5  ninths,  |, 

5  sixths,  I ;          6  sevenths,  f ;          5  eighths,  | ;          6  ninths,  f . 

Numbers  which  express  one  of  the  equal  parts  of  an 
integer ;  as  -J^,  4,  i,  7^,  3V  ;  ov  which  express  an  equal  part 
of  two  or  more  integers,  or  two  or  more  equal  parts  of  a 
one ;  as,  |,  f ,  /y,  -^Jq  ;  form  a  class  of  numbers  called  Frac- 
tions.    Hence, 


ISO  FK  ACTIONS. 

301.  A  J^raction  is  a  number  which  expresses  one  or 
more  of  the  equal  parts  into  which  a  one  or  any  other 
integer  is  divided. 

A  fraction  is  expressed  by  two  numbers,  written  one 
under  the  other,  with  a  horizontal  line  between  them. 

302.  The  2'erms  of  a  fraction  are  the  two  numbers  used 
to  express  it.     Thus,  the  terms  cf  f  are  5  and  7. 

303.  The  ^e7iominat07*  of  a  fraction  is  that  term 
which  expresses  the  number  of  equal  parts  into  which  gnd 
is  divided  ;  ifc  is  written  below  the  horizontal  Hne.     And 

304.  The  Mi7neraior  is  that  term  which  expresses  the 
number  of  equal  x^arts  indicated  by  the  fraction  ;  it  is  writ- 
ten above  the  line.  Thus,  in  the  fraction  |,  the  5  is  the 
denominator,  and  expresses  that  a  unit  or  1  is  divided  into 
5  equal  parts,  and  4  is  the  numerator,  and  expresses  that 
4  of  the  equal  parts  (fifths)  are  indicated  by  the  fraction. 

305.  The  Reciprocal  of  a  number  is  the  quotient  of 
a  one  divided  by  that  number.  Thus,  the  reciprocal  of  7  is 
4,  of  $13  is  %^.,  of  25  bu.  is  ^^  bu. 

306.  A  ^racHo7ial  Ujiit  is  one  of  the  units  of  the 
numerator.  Its  value  is  expressed  by  the  reciprocal  of  the 
denominator. 

Note. — A  fractional  unit  may  be  either  abstract  or  concrete.  Thus,  tlic 
fractional  unit  of  f  is  \,  of  $|  is  %\,  of  f  ft.  is  \  ft.,  of  \  lb.  is  \  lb. 

307.  The  value  of  a  fraction  depends  upon  the  relative 
values  of  its  numerator  and  denominator. 

I.  Wlien  the  numerator  and  denominator  are  equal,  the  value 
of  the  fraction  is  1 ;  because  as  many  fractional  units  are 
expressed  as  equal  an  integral  unit  or  1.     Thus,  ^,  -]f,  ||g. 

n.  Wlien  the  numerator  is  less  than  the  denominator,  the 
value  of  the  fraction  is  less  than  1 ;  because  a  less  number 
of  fractional  units  is  expressed  than  equal  an  integral  unit 

nr  1        Tlinc!    3    _7      15      158 

m.    When  the  numerator  is  greater  than  the  denominator, 


NOTATION    AND    PRINCIPLES, 


181 


the  value  of  the  fraction  is  more  than  1 ;  because  a  greater 
number  of  fractional  units  is  expressed  than  equal  an 
integral  unit  or  1.     Thus,  -^,  ^-  ^"^    ^-'^■'- 

808.  A  Proper  I^r ac- 
tion is  a  fraction  whose 
value  is  less  than  1 ;  as  J, 

1  0»    6'    1 G* 

309.  An  Im^)  roper 
^i^aciion  is  a  fraction 
whose  value  equals  or  ex- 
ceeds 1 ;  as  I,  jg,i  ii,  ff. 

310.  A  Mixed  JViim- 
ber  is  a  number  expressed 
by  an  integer  and  a  deci- 
mal, or  an  integer  and  a 
fraction  ;  as  3.7,  21.4,  9.35; 
3i,  llf ,  1415. 

Note. — In  reading  a  mixed  num- 
ber, and  belongs  between  the  in-   j— . 
teger  and  the  fraetion  or  decimal.  □— ' 

311.  Similar  JF^rac- 

li07is  are  fractions  that 
have  a  common  fractional 
unit ;  as  1l,  %  ;  %  %,  %. 

312.  !2)is  similar 
fractions  are  fractions 
that  have  different  frac- 
tional units  ;  as  f ,  4 ;  f ,  f ,  {7. 

313.  If  we  multiply  the 
numerator  of  \  by  2,  we  obtain  |.  The  fi-actional  unit 
in  \  and  §  is  the  same.  If  we  multiply  the  numeratoi* 
of  \  by  3,  we  obtain  |, — a  number  of  3  times  as  many  frac- 
tional units  as  i,  each  unit  of  both  fractions  being  of  the 
same  value.     That  is,  ^""^=1,  |''-  =  f,  |''^=|,  etc.     Hence, 

Multiplying  the  numerator  multiplies  the  fraction.  (See  272.) 


MPROPER    FHRCTIONS 


182  FRACTIONS. 

314*  If  we  divide  the  numerator  of  |  by  2,  we  obtain  |,  a 
number  of  one  half  as  many  fractional  units  as  f,  each 
unit  of  both  fractions  being  of  the  same  value.  So,  also,  if 
we  divide  the  numerator  of  f  by  3,  we  obtain  f.      That  is, 

2-5-2—1      2-^2—1      6^3—2     p+p         HPTIfP 

B      — 8>  4      — 4>  "7      — "7»  ^^^'     -tience. 

Dividing  the  numerator  divides  the  fraction.     (See  273.) 

315.  If  we  multiply  the  denominator  of  |  by  2,  we  ob- 
tain |,  a  fraction  of  the  same  number  of  fractional  units  as 
^,  each  unit  of  the  |  being  one  half  the  value  of  a  unit  of 
the  |.  So,  also,  if  we  multiply  the  denominator  of  A  by  2, 
we  obtain  {.    That  is,  ^xo  =  |,  ix2  =  -h  |x3  =  r5J  etc.   Hence, 

Multiplying  the  denommator  divides  the  fraction.    (See  274.) 

316.  If  we  divide  the  denominator  of  |  by  2,  we  obtain 
I,  the  number  of  fractional  units  in  the  |  being  the  same  as 
in  the  |,  while  the  value  of  each  unit  is  2  times  as  great. 
So,  also,  if  we  divide  the  denominator  of  J  by  3,  we  obtain 
i.     That  is,  1^2  =-|>  1^2 =i  T o-^a  =  i  etc.     Hence, 

Dividing  the  denominator  multiplies  the  fraction.    (See  275.) 

317.  If  we  multiply  both  terms  of  \  by  2,  we  obtain  §. 
The  number  of  fractional  units  in  §  is  2  times  as  many  as  in 
\y  but  the  value  of  each  unit  is  only  one  half  as  much.  In 
other  words,  the  fraction  \  is  multipHed  by  2  by  multiply- 
ing its  numerator  by  2,  and  the  result  (f )  is  divided  by  2 
by  multiplying  its  denominator  by   2.      That  is,  ^x2=4> 

1X2  —  2      3X4 — 12     pfp         TTA-nr»o 

4X2  —  8»  "7x4  —  3s>  etc.     xience, 

Multiplying  both  terms  of  a  fraction  by  the  same  number 
does  not  change  its  value.     (See  276.) 

318.  If  we  divide  both  terms  of  |  by  2,  we  obtain  f . 
.The  number  of  fractional  units  in  |  is  one  half  as  many  as 
in  |,  but  the  value  of  each  unit  is  2  times  as  much.  In 
other  words,  the  fraction  |  is  divided  by  2  by  dividing  its 
numerator  by  2,  and  the  result  (g)  is  multiplied  by  2  by 

4-^2 2     2^-2 1 

H-^2  —  4'   4+2  —  2 J 


dividing  its  denominator  by  2.     That  is,  |ii  — ^   2+2  —  1 


|,  etc.     Hence, 


EEDUCTIONS.  183 

Dividing  both  terms  of  a  fraction  bij  the  same  number  does 
not  change  its  value.     (See  277.) 

319.  The  deductions  in  the  last  six  articles  are  the 

General  Principles  of  I^ractions, 

I.  A  fraction  is  multiplied  by  multiplying  its  numerator  or 
dividing  its  denominator. 

n.  A  fraction  is  divided  by  dividing  its  numerator  or  mul- 
tiplying its  denominator. 

III.  The  value  of  a  fraction  is  not  changed  by  either  mul- 
tiplying or  dividing  both  terms  by  the  same  number,    see  Manual. 

820.  In  integers,  decimals,  and  compound  numbers  the 
successive  orders  of  units  increase  and  decrease  by  fixed 
scales.  In  fractions  the  scales  (that  is,  the  number  of  frac- 
tional units  required  to  equal  an  integral  unit)  vary  with 
every  change  of  the  denominator.  This  feature  of  fractions 
gives  rise  to  the  principal  difference  between  computations 
in  fractions  and  integers,  decimals,  and  compound  numbers. 


SECTION  II. 

C^SE     I. 
Fractions  to  Lo-west  Terms. 

321  •  A  fraction  is  in  its  Lowest  Terms,  when  its  numera- 
tor and  denominator  are  prime  to  each  other  ;  as  |,  f ,  |, 

23 

36' 

When  the  terms  of  a  fraction  are  not  prime  to  each 
other,  they  have  some  common  factor. 

Ex.  Reduce  ^f  to  its  lowest  terms. 

Explanation. — Since  the  value  of  a  frac-         first  solution. 
tion  is  n6t  changed  by  dividing  both  terms  ||— /^=| 


184  FRACTIONS. 

by  the  same  number  (see  319,  III.),  we  re- 
duce ^f  to  lower  terms,   by  dividing  its       skcond  solution. 
terms  by  the  common  factor  2,  (If  =  13)  5  ij=T 

and  the  result,  -^%,  we  reduce  to  still  lower 
terms,  by  dividing  its  terms  by  the  common  factor  3, 
(-»2— I)  as  shown  in  the  Fu'st  Solution.  Since  the  terms  3 
and  4  are  prime  to  each  other,  |  must  be  the  lowest  terms 
of  the  fraction  4f  ;  and  coiisequently  |  is  the  result  re- 
quired. Or,  we  can  reduce  ^f  to  its  lowest  terms  at  one 
operation,  by  dividing  both  terms  by  their  greatest  com- 
mon divisor,  6,  as  shown  in  the  Second  Solution. 

phoblems. 

1.  Reduce  the  fraction  |^f  to  its  lowest  terms.  s. 

2.  Reduce  ^  and  f  i  to  their  lowest  terms.  ^^  4^ 

3.  Reduce  ^,  -j^^,  and  -^^  to  their  lowest  terms. 

4.  In  what  lower  terms  can  the  value  of  ff  be  expressed  ? 

In  four  different  fractions. 

5.  Wliat  are  the  lowest  terms  of  ^  and  f  f  ? 

6.  What  are  the  lowest  terms  of  the  fractions  ff ,  f^,  -^j^  and 

CJLSE    II. 
Fractions  to  Given  Denominators. 

S22,  Ex.  Reduce  |  to  a  fraction  having  42  for  a  denom- 
inator. 

Explanation. — Since  the  value  of  a  fraction  solution. 

is  not  changed  by  multiplying  both  terms  by      i?  L^' 
the  same  number  (see  319,  TIL),  we  multiply        6 
both  terms  of  f  by  an  integer  that  will  give      ^xg  —  sq 
42  for  a  new  denominator.     We  find  this  in-  ^t     *     .^1_ ,,  ^ 
teger  by  dividing  42,  the  required  denomi-  '  ''^~^^ 

nator,  by  7,  the  denominator  of  •].  Then,  multiplying 
both  terms  of  f  by  6,  the  integer  thus  found,  we  have  |^, 
the  fi'action  required.  s 


REDUCTIONS.  185 

PJt  OBLBMS. 

7.  Eeduce  f  to  sixteentlis.  ^|, 

8.  Reduce  f  to  tenths,  and  to  twenty-fifths. 

9.  Reduce  |-  to  a  fraction  having  63  for  a  denominator.  /  '" 

10.  Reduce  -|  to  54ths,  and  ^V  to  84ths.  A|^  A|.. 

11.  Reduce  f,  f,  and  \  to  sixtieths. 

12.  In  ■jfg-  are  how  many  twenty-sixths,  how  many  sixty-fifths, 
and  how  many  ninety-firsts  ?  j?     ±i>.  z± 

C^SEJ    III. 
Dissimilar  Fractions  to  Similar  Fractions. 

•     323.  Ex.  1.  Eeduce  \  and  |  to  similar  fractions. 

Explanation. — Fourths    can  not   be  solution, 

reduced  to  fifths,  nor  fifths  to  fourths.  |^^|~/j^ 

But  since  4  times  5  =  20,  we  reduce  \  §-^f=-^-§- 

to  twentieths  by  multiplying  iLs  terms  tt^„^^  -f   a—  5    -i2 
by  5  ;  and  since  5  times  4  =  20,  we  re- 
duce I  to  twentieths  by  multiplying  its  terms  by  4. 

Ex.  2.  Eeduce  f ,  |,  and  f  to  similar  fractions. 

Explanation, — Since  the  solution, 

product  of  the  denomina-  f^|^^— _'y)^ 

tors,   3   times  5   times   7,  a^|^7— _s_a, 

=  105,    we    may    reduce  f^|^|=:--?-«j- 

these  fractions  to  105ths.       t-t^^^p  2   a    0-^0      sa      90 
inis  we  do  by  muitipiymg 

the  terms  of  the  first  fraction,  §,  by  5  and  7 ;  the  terms  of 
the  second,  |,  by  3  and  7 ;  and  the  terms  of  the  third,  f ,  by 
3  and  5.  That  is,  wo  multiply  the  terms  of  each  fraction 
by  the  denominators  of  the  other  fractions. 

Erom  these  examples  it  will  be  seen  that 
The  denominator  of  the  similar  fractions  is  a  common  mul- 
tiple of  the  denominators  of  all  the  given  fractions. 

Notes. — 1.  Fractions  having  like  denominators  are  said  to  have  a  Com- 
mon De7iominator . 

2.  Reducing  dissimilar  to  similar  fractions  is  sometimes  called  reducing 
fractions  to  equivalent  fractions  having  a  common  denominator. 


186  FRACTIONS. 

PJtOBLEMS. 

13.  Reduce  ^-  and  f  to  similar  fractions.  s   a 

14.  Reduce  f  and  f  to  similar  fractions. 

15.  What  similar  fractions  are  equal  to  f  and  ^  ? 
IG.  What  similar  fractions  are  equal  to  f ,  f ,  and  |  ? 

17.  Reduce  ^  and  -^j  to  similar  fractions.  or^^    is^ 

18.  Reduce  f ,  f ,  f ,  and  f  to  similar  fractions. 

19.  Reduce  -^^  f,  f,  and  -^  to  equivalent  fractions  having  a 
common  denominator.  st^ko   jl.ioo   44 ar  gjinA 

20.  Reduce  |,  f ,  and  ^  to  equivalent  fractions  having  a  com- 
mon denominator. 

Dissimilar  Fractions  to  Least  Similar  Fractions. 

324.  Since,  in  reducing  dissimilar  to  similar  fractions,  the 
common  denominator  must  be  a  common  multiple  of  the 
denominators  of  all  the  given  fractions  (see  323),  it  follows 
that 

The  common  denominator  of  least  similar  fractions  must  be 
the  least  common  multiple  of  the  denominators  of  all  the  given 
fractions. 

325.  Ex.  Keduce  |,  |,  and  ^ 
to  least  similar  fractions. 

Explanation. — We  first  find 
the  least  common  multiple  of  all 
the  given  denominators  6,  4,  9, 
to  be  36  (see  299).  Since  36  is  the 
common  denominator  of  the 
least  similar  fractions  that  are 
equal  to  the  given  fractions  |, 
|,  and  ^,  we  reduce  each  of 
these  fractions  to  36ths  by  Case 
II.     (See  322.)  ] 


6= 

Jf^ 

-.2x2 

9= 

3x3 

3x 

3x2x 

2=36 

36 

[6 

^u 

36  19 

6 

9 

4 

KS 

=M 

4X9 

=A 

AX-4 

yxji. 

=U 

ce, 

•hh 

i=M. 

/^,  U' 

KEDUCTIONS.  187 

Notes.— 1.  The  fractional  units  of  dissimilar  fractions  are  unlike,  but 
the  fractional  unit  of  their  equivalent  similar  fractions  is  common.  (See 
311.) 

2.  The  fractional  unit  of  least  similar  fractions  is  the  greatest  fractional 
unit  common  to  the  given  dissimilar  fractions.      See  Manual. 

PJlOBI,E3lS, 

21.  Reduce  -|  and  -|  to  least  similar  fractions,  ^|-^  i.|;, 

22.  Reduce  -^  and  ^V  to  least  similar  fractions.  _4_     r 

23.  What  are  the  least  similar  fractions  equal  to  |  and  f  ? 

24.  What  least  similar  fractions  are  equal  to  |,  ^,  y'^,  and  |  ? 

25.  Reduce  -^j,  |,  and  f  to  least  similar  fractions. 

/   26.  Reduce  ^i,  f,  I,  If,  and  ^  to  least  similar  fractions. 

sfi    an    55    .9S    y.7 

X5?  :4  3^J  J5>  7.3^J  J3' 

27.  AYhat  is  the  fractional  unit  of  the  least  similar  fractions  to 
which  f ,  f ,  yV?  and  y^^F  ^an  be  reduced  ?  _       i^^-. 

Improper  Fractions  to  Integers  or  Mixed  Numbers. 

326.  Ex.  1.  How  many  ones  in  ^^^-  ? 

Explanation. — Since   every  7  sev-  solution. 

enths  are  1,  21  sevenths  are  as  many      ^  sevenths.  1 7  sevenths. 
I's  as  the  number  of  times  7  sevenths        3 
are  contained  in  21  sevenths,  which 
is  3  times. 

Ex.  2.  Find  the  value  of  the  improper  fraction  -^^-. 

Explanation. — Since  every  5  fifths  are  soLUTioii. 

1,  17  fifths  are  asmany  I's  as  the  number  ^yy'i^'-  \^5  fifths. 
of  times  5  fifths  are  contained  in  17  fifths.  S§- 
The  quotient  figure  is  3  ;  and  since  the 
remainder  is  always  like  the  dividend  (see  109,  Yin.),  and 
the  dividend  is  fifths,  the  remainder  2,  is  2  fifths  or  f. 
Writing  the  |  at  the  right  of  the  quotient  figure  3,  we  have 
3|,  the  value  required. 

Or,  we  may  regard  the  numerator  and  denominator  as 
dividend  and  divisor,  and  both  concrete  numbers  (fifths). 


188 


FRACTIONS. 


Then,  the  quotient  figure  is  3,  and  the  remainder,  2,  is  a  less 

number  to  be  divided  by  a  greater,  5  ;  and  the  result  is  | 

(see  301).     Writing  the  |  at  the  right  of  the  3,  we  have 

the  abstract  number  3?,  as  before.     (See  109,  III.) 

Note. — Any  dividend  may  be  written  as  tlie  numerator,  and  the  divisor 
as  the  denominator  of  a  fraction. 

Fit  OBJLEMS. 

28.  How  many  apples  are  ^  apples  ? 

29.  In  -^  miles  are  how  many  miles  ? 

30.  Reduce  the  improper  fraction  -^  to  a  mixed  number.    .^|. 

31.  Reduce  ■^-  to  an  integer.  Ih 

32.  How  many  yards  are  ^f  yd.  ? 

33.  Find  the  integer  or  mixed  number  equal  to  each  of  the  im- 
proper fractions  %^^,  -Vs"  ^la.,  V/,  ^¥-  ft-,  ^fP  cu.  yd.,  ^^  lb. 

34.  29  quarter-dollars  are  how  many  dollars  ? 

35.  Reduce  \^-,  ^^  ^-^  ^-,  and  ^-^  to  integers  or  mixed  num- 

'43 


bers. 


6^- 


),  51-,  me. 


ca.se:  "vi 


Integers  or  Mised  Numbers  to  Improper  Fractions. 


SOLUTION. 


FULL  SOLUTION. 

7  seventh^,. 


327.  Ex.  1.  Reduce  the  integer  8  to 
fifths. 

Explanation. — Since  1  is  5  fifths,  8 
are  8  times  5  fifths,  or  40  fifths. 

Ex.  2.  Reduce  the  mixed 
number  4|  to  an  improper 
fraction. 

Explanation.  —  Since  1  is  7 
sevenths,  4  are  4  times  7  sev- 
enths, or  28  sevenths  ;  and  28 
sevenths  +  3  sevenths  are  31 
sevenths. 

The  reduction  of  4|  to  sevenths  is  similar  to  the  reduc- 
tion of  a  compound  number  of  two  denominations  to  the 


J 

28  sevenths. 
3  sevenths. 

31  sevenths. 


_8 

JfOfftJ^' 
Hence,  8  =  -^/. 

COMMON  SOLUTION. 

28  +  3=31 


Hence,  #=V- 


REDUCTIONS.  189 

lower  denomination.  Thus,  the  4  ones  corresponds  to  the 
higher  denomination,  and  the  3  sevenths  to  the  lower.  In 
the  second  or  Common  Solution  we  reduce  the  4  ones  to 
sevenths  and  add  the  given  3  sevenths,  in  the  same  manner 
as  we  would  reduce  4  wk.  3  da.  to  days.     (See  223.) 

1*JR  OBI.E3IS. 

36.  Reduce  12  to  sevenths,  and  13  to  ninths.  ^^         &A^  ±-t7^, 

37.  In  5|  are  how  many  eighths  ?  .^?.. 

38.  Reduce  19f  to  an  improper  fraction. 

39.  Reduce  b-^^  and  43^  to  improper  fractions.        |7,  ^M^- 

40.  What  improper  fractions  are  equal  to  15fV  and  17-|  ? 

41.  Change  14||,  13/g^,  and  llf  to  improper  fractions. 

42.  In  365^  days  there  are  how  many  fourths  of  a  day  ?    ■ia£,±  da, 

328.  Brief  directions  for  performing  the  processes  in  the 
preceding  six  Cases  form  the 

Steles  for  deductions  of  J^ractlons, 

I.  Fractions  to  lowest  terms. 
Cancel  all  the  factors  common  to  both  tenns. 

n.  Fractions  to  given  denominators. 
Divide  the  given  denominator  by  the  denominator  of  the 
fraction,  and  multiply  both  terms  of  the /inaction  by  the  quotient. 

m.  Dissimilar  to*  similar  fractions. 
Multiply  both  terms  of  each  fraction  by  the  denominators  of 
all  the  other  fractions. 

IV.  Dissimilar  to  least  similar  fractions. 

1.  For  the  least  common  denominator,  find  the  least  common 
multiple  of  all  the  denominators. 

2.  For  each  new  numerator,  divide  the  least  common  multi- 
ple by  the  denominator  of  each  fraction,  and  multiply  the  nume- 
rator by  the  quotient. 

V.  Improper  fractions  to  integers  or  mixed  numbers. 
Divide  the  numerator  by  the  denominator. 


190  FKACTIONS. 

VI.  Integers  or  mixed  numbers  to  improper  fractions. 

1.  Mulliply  the  integer  by  the  denominator,  and  if  there  he  a 
numerator,  add  it  to  the  product. 

2.  Write  this  result  and  the  given  denominator  for  the  terms 
of  the  required  fraction. 


PJtOB  ZEMS. 

43.  To  what  lower  terms  can  -^^  be  reduced  ?   i-j:^,  /  5    2±^  ._7_^ 

44.  How  many  one  hundred  fifty-thirds  are  equal  to  eleven  seven- 
teenths ?  JL?_ 

^53' 

45.  Reduce  y^,  ^,  and  ^  to  equivalent  fractions  having  a  com- 
mon denominator. 

46.  What  similar  fractions  arc  equal  to  |-,  I,  and  -^-^  ? 

47.  Reduce  -^^^  |^f ,  -5^,  and  -^  to  least  similar  fractions. 

48.  Reduce  -^,  ^f^,  i^,  ip-,  and  ^^  to  integers  or  mixed 
numbers. 

49.  Reduce  59  to  a  fraction  having  59  for  a  denominator.     Re- 
duce it  to  9ths.  -H~i  H~' 

50.  What  least  similar  fractions  are  equal  to  \^  f ,  -^^,  -^^  and  ^\  ? 

51.  What  is  the  greatest  common  fractional  unit  of  H,  |-|,  -j'^, 
andfi?  7^^. 

53.  Find  the  lowest  terms  of  ^W^,  fii,  and  ^f^. 

53.  Change  -^-^  to  ninety-fifths,  to  one  hundred  seventy-firsts,  and 
to  two  hundred  ninths.  ^|,  //^,  //g-. 

54.  Reduce  the  fractions  |-|,  f  ^  y^?  t>  ^^^  H  *^  eighty-fourths. 

55.  What  are  the  lowest  terms  of -j^V??  fli»  tHHti  and  ^^^  ? 

56.  Wbat  similar  fractions  are  equal  to  -^^  and  -^  ? 

57.  Reduce  |-,  |,  I'V,  and  ^V  to  similar  fractions. 

58.  Reduce  f ,  |-,  f ,  |,  f ,  f ,  and  f  to  fractions  having  a  common 
fractional  unit. 

59.  What  least  similar  fractions  are  equal  to  y^o,  I,  tV,  and  ^^  ? 

60.  Reduce  lOOOyoW  ^^^  ^^yV  to  improper  fractions. 

61.  What  improper  fractions  are  equal  to  67^^  and  133f  ? 
63.  Reduce  ^^^,  -^^  ■^,  and  |^  to  least  similar  fractions.  . 

Their  greatest  common  fractional  unit  is  j-^^. 


ADDITION    AND    SUBTRACTION.  191 

SECTION   HI. 

329.  Since  only  like  orders  of  units  can  be  added  one  to 
another  (see  39,  II.),  or  subtracted  one  from  another  (see 
^%,  II.),  and  dissimilar  fractions  are  of  unhke  orders  of 
units,  it  follows  that  they  must  be  reduced  to  similar  frac- 
tions, (that  is,  to  the  same  fractional  unit),  before  they  can 
be  added  or  subtracted. 

C^SE     I. 
All  the  Given  Numbers  Fractions. 

330.  Ex.  1.  What  is  the  sum  of  |,  |,  and  \  ? 

Explanation. — The  giv-  fikst  solutioit. 

en  fractions  being  dissimi-  i  +  §  +  i^=H  +  U+U  =  H=' Hi 
lar,  we  first  reduce  them 

'  _  SECOND  SOLUTIOrr, 

to  the  similar  fractions  ;?.  j_ .2  _|_  i — ^3+2^+^0  _ _?_»  —  74.5 
|-§,  §4,  and  Jg.     Since  the 

parts  of  these  similar  fractions  are  all  of  the  same  kind  or 
denomination  (sixtieths),  and  since  the  numerators  express 
the  numbers  of  the  parts,  we  add  the  similar  fractions  by 
adding  their  numerators,  45  +  24  +  10  =  79  ;  and  since  the 
fractional  unit  of  the  parts  is  g^^,  we  write  the  denominator, 
60,  under  the  79,  making  l^.  Then,  reducing  the  Jg  ^^  ^ 
mixed  number,  we  have  1J§,  the  result  required. 

Ex.  2.  Subtract  |  from  |. 

Explanation. — The    given    fractions  ^^^^t  soltttion. 

being  dissimilar,  we  first  reduce  them  '8~§-=io~jo—ii 
to  the   similar   fractions   |4   and  M. 

4  0  4  0  SECOND  BOLTTTION. 

Since   all  the   parts   of   these   similar      7^—3^—2Jf-^sj.~±j. 
fractions  are  of  the  same  kind  or  de- 
nommation  (fortieths),  we  subtract  24  fortieths  from  35 
fortieths,  and  the  difference,  |  J,  is  the  result  required. 


192  FRACTIONS. 

In  reducing  dissimilar  to  similar  fractions,  the  common 
denominator  need  be  written  but  once,  and  the  several 
numerators  may  be  written  above  it,  connected  by  the  ap- 
propriate signs,  as  shown  in  the  Second  Solution  of  each 
of  the  two  preceding  examples. 

From  these  examples  we  learn  that 

I.  The  numerators  of  similar  fractions  only  can  be  added  or 
subtracted;  and 

H,  The  common  denominator  is  icritten  under  the  sum  or 
difference. 

I*ROBLEMS.' 

1.  What  is  the  sum  of  i  and  -|  ?  ±i 

2.  What  is  the  sum  ot  4  and  4  ?  4±, 

3.  What  is  the  difference  between  -|  and  f  ?  /^  • 

4.  From  -^^  subtract  f .  |2. 

5.  William  gatliered  |-  bu.  of  butternuts  one  day,  and  f  bu.  the 
next.     How  many  did  he  gather  in  the  two  days  ? 

G.  From  |  yd.  of  velvet  a  lady  used  -|-  yd.  How  much  velvet 
had  she  left  ? 

7.  A  Michigan  farmer  made  ^  T.  of  maple  sugar,  and  £)old  -|  T. 
How  much  sugar  did  he  keep  ?  ^^  T. 

8.  The  tide  rose  |  ft.  one  hour,  -^f  ft.  the  next  hour,  and  |  ft.  the 
third  hour.     How  much  did  it  rise  in  the  three  hours  ?      S-U.  ft. 

C^SE     II. 
Any  of  the  Given  Numbers  Mixed  Numbers. 

331.  Ex.  1.  What  is  the  sum  of  5f,  -^,  6^,  and  11  ? 

Explanation. — We  wi'ite  the  given  numbers 
in  columns,  integers  under  integers,  and  frac-  ^^°^"°%^ 
tions  under  fractions.  Eeducing  the  frac-  i~  ij 
tional  parts  to  similar  fractions,  we  have  5|  qi  —  ^sj. 
=  ^h  i  =  ~4h  and  6A  ^  6f i.  Adding  the  i/  =.  ii'''' 
fractions^  we  have  ||  or  1||.    We  write  the  5S 

f  ^  in  the  result,  and  add  the  1  with  the  given 


ADDITION    AND     SUBTKACTION.  193 

integers.     23,  tlie  sum  of  all  the  integers,  written  before 
the  |A,  gives  23|A,  the  required  sum. 

Ex.  2.  From  7^  subtract  3§. 

Explanation. — ^We  write  the  subtrahend  un-        solutiox. 
der  the  minuend,   and  reduce  the  fractional      ^j  =  7';/^ 
parts  to  similar  fractions.    Since  |§  can  not  be      "^^  ~  "^sf 
subtracted  from  /g,  and  since  the  difference  o§§- 

will  not  be  affected  by  adding  the  same  number 
to  both  minuend  and  subtrahend  (see  52,  III.),  we  add  || 
(=  1)  to  the  3%  of  the  minuend, ^nd  1  (=  ||)  to  the  3  of 
the  subtrahend.  We  then  subtract  |g  from  ||  {=m^), 
and  4  from  7,  writing  the  results,  1 1  and  3,  as  the  fractional 
and  integral  parts  of  the  remainder.  The  result,  3§|,  is  the 
remainder  required.      See  Manual. 

When  any  of  the  given  numbers  are  mixed  numbers,  we 
may 

Regard  the  fraction?,  as  lower,  and  the  integers  as  higher 
denominations,  and  add  and  subtract  as  in  compound  7iumbers. 

PROBLEMS. 

9.  What  is  the  sum  of  4|  and  3^  ?  8^'^. 

10.  From  6yV  subtract  3f.  4j\. 

11.  A  lady  bought  15|-  yd.  of  delaine,  llf  yd.  of  calico,  and 
4f  yd.  of  merino.     How  many  yards  of  dress  goods  did  she  buy  ? 

13.  I  bought  13f  cd.  of  wood,  and  at  the  end  of  a  year,  had 
1-jV  cd.  left.     How  much  had  I  used  ?  iii-|  al 

13.  A  mechanic  spent  $9^'o  ^^^  ^1^  week's  wages,  and  had  $3f 
left.     What  was  the  amount  of  his  wages  ? 

14.  A  merchant  sold  a  pair  of  fur  gloves  for  $3^,  upon  which 
his  profit  was  Iff.    What  was  the  first  cost  of  the  gloves  ? 

15.  My  farm  consists  of  five  fields  that  contain  respectively 
13^^  A.,  15f  A.,  13^^  A.,  11^  A.,  and  14|  A.  How  many  acres 
in  my  farm  ?  ^^?^V* 

9 


194  FRACTIONS- 

332.  Upon  the  principles  deduced  in  330,  331,   is  based 
the 

'Utile  for  Addition  and  Subtraction  of  J^ractions, 

I.  Reduce  dissimilar  to  similar  fractions. 
n.  Add  or  subtract  the  numerators,  and  under  the  result 
write  the  common  denominator. 

Notes. — 1.  If  the  given  fractions  are  reduced  to  least  similar  fractions, 
the  numerators  to  be  added  or  subtracted  will  be  the  smallest  numbers 
possible. 

2.  In  all  final  results  reduce  fractions  to  lowest  terms,  and  improper  frac- 
tions to  integers  or  mixed  numbers. 

JPM  OB  Z  JEMS. 

16.  What  is  the  sum  of  f  and  ^  ?  as^ 

17.  From  I  subtract  ^.  iz, 

18.  George  paid  $|-  for  a  pair  of  skates,  and  $^  for  straps. 
What  was  the  whole  cost  ? 

19.  From  7^^  subtract  6f .  -<|.. 

20.  The  parts  are  4f ,  S^-,  3|,  4^,  and  l^^.     What  is  their  sum  ? 

21.  Mary  had  $^,  but  she  spent  $f  for  a  ribbon.  How  much 
money  has  she  left  ?  ^jg-* 

22.  What  is  the  sum  of  |,  f ,  and  f  ? 

23.  From  f  subtract  1^.  §f. 

24.  A  lady  purchased  a  shawl  for  $8f ,  and  gave  the  merchant  a 
10-dollar  bill.    How  much  change  should  she  receive  ? 

25.  If  a  family  bum  f  T.  of  coal  in  Dec,  |  T.  in  Jan.,  and  ^f  T. 
in  Feb.,  how  much  do  they  burn  in  the  three  months  ? 

26.  How  much  greater  is  ^  than  -^^  ?  ^y. 

27.  A  merchant  sold  a  lace  collar  for  $||-,  that  had  cost  him  |^. 
How  much  was  his  profit  ?  ^||. 

28.  A  contractor  having  a  contract  to  build  2d^  mi.  of  rail- 
road, has  completed  14yV  nai.    How  much  has  he  yet  to  build  ?     , 

29.  From  32tV  subtract  ^.  ^^iuT*  ^^ 

30.  Add  fl,  i»  1^,  -h^  and  ^.  8um,  2^. 


MULTIPLICATION.  195 

31.  A  stone-mason  in  building  a  wall,  used  f  cd.  of  stone  one 
day,  -^^  cd.  the  second  day,  -^  cd.  the  third  day,  and  y*j  cd.  the 
fourth  day.    How  much  stone  did  he  use  in  the  four  days  ? 

32.  A  founder  used  ^  T.  of  iron  in  making  f  T.  of  castings. 
How  much  was  the  waste  ?  -^^  T, 

33.  If  the  less  of  two  numbers  is  7^,  and  the  greater  is  37^, 
what  is  the  difference  ?  -^^f  !• 

34.  Find  the  sum  of  391f ,  19^,  4tV,  57^,  and  -i^.        473i-§. 

35.  -^^  is  how  much  greater  than  -^^  ? 

36.  What  is  the  distance  round  a  farm  |-  mi.  long  and  ^^  mi. 
wide  ?  ^1^^  mi. 

37.  What  is  the  sum  of  f?  and  ^  ?    What  is  their  difference  ? 

Sum,  1/^2  '  differerwe,  //g-. 

38.  The  minuend  is  11^,  and  the  subtrahend  5^.  What  is  the 
remainder  ? 

39.  A  cake  of  ice  1|4-  ft-  thick  floats  with  -^^^^i.  of  its  thickness 
above  the  water.    What  thickness  of  the  ice  is  under  water  ? 

1-?-^-  ft 

■^-i  OS  J'" 

40.  A  farmer  sold  13^  T.  of  his  hay  crop,  put  llf  T.  into  his 
bam,  and  stacked  9^  T.    How  much  hay  did  he  raise  ?    SJ/.^^^  T, 


SECTION      IV. 
MZrZ  TITJDIC;^  TZOJV. 

C^SE  I. 
One  Factor  a  Fraction. 
333.  Ex.  1.  Multiply  j%  by  5. 

Explanation. — ^In  the  First  Solu-  ^^^  soLtj-now. 
tion  we  have  multiplied  8,  the  numer-       7J     —TJ~^i^—^§ 

ator  of  the  fraction,  by  5,  and  in  the  „„„„„„  „ ^„ 

'       "^        '  SECOND  SOLUTION. 

Second  Solution  we    have  divided  _^^^  __|_^|: 

15,  the  denominator,  by  5  (see  319, 1.) 

The  results  in  the  two  solutions  are  the  same. 


196 


FRACTIONS. 


FIKST  SOLUTION. 


tV 


/:,X5=ff  =  ^/^: 


SECOND  SOLUTION. 


n 


7  y  _.7   —  .9  5  - 


■■^7%- 


Ex.  2.  Multiply  7  by  /j,  or  find  j%  of  7. 

Explanation. — To  multiply  a  num- 
ber by  5  is  to  find  5  times  the  num- 
ber ;  to  multiply  it  by  1  is  to  find  1 
time  the  number  ;  to  multiply  it  by 
j\  is  to  find  ^5  of  it  ;  and  to  multi- 
ply it  by  ^4  is  to  find  5  times  j\  or 
j\  of  it.  In  the  First  Solution  we 
divide  7  by  14,  and  obtain  j\,  which 
is  j\  of  7  ;  and  we  then  multiply 
this  result  by  5,  and  obtain  f  f ,  or  2i 

Since  7  x  -f^  =  f 5  x  7  (see  80,  V.),  in  the  Second  Solution 
we  multiply  7  and  y\  together,  in  the  manner  explained  in 
Ex.  1. 

1.  Multiply  ^^  by  8,  or  find  8  times  ^. 

2.  How  much  is  1 3  times  ^  ? 

3.  At  $1  a  yard,  how  much  will  ^  yards  of  alpaca  cost  ? 

4.  Multiply  18  by  -^\,  or  find  -^V  of  18. 


Or 


which  is  j\  of  7.. 


^A- 


S4i. 


5.  How  much  is  A  of  14  miles  ? 


6.  What  is  the  product  of  19  and  -^1    Of  31  and  ||  ?     Of 
and  34?  J,-f,-,  19^,  mh 

7.  How  much  will  12  gal.  of  kerosene  cost,  at  $|f  per  gal.  ? 

8.  A  farmer  bought  f  bu.  of  grass  seed  @ 
it  cost  him  ? 


How  much  did 


Cj^SE     II. 
Both  Factors  Fractions. 

331.  Ex.  1.  Multiply  |  by  |,  or  find  I  of  §. 

Explanation. — |  of  any  number  is  3  times 
as  much  as  |  of  it,  and  |  of  it  is  found  by 
dividing  it  by  4.  In  the  First  Solution  we 
multiply  the  denominator  of  |  by  4  to  find  \ 
of  I  (see  319,  n.).  We  then  multiply  the 
numerator  of  the  result,  5^,  by  3,  to  find  3 


FIKST  SOLUTION 

■t 

SECOND 

SOLUTION. 

MULTIPLICATION.  197 

times  ^,  or  |,  of  |.     This  result,  if,  =  |,  the        -^^^^^^  sm^ution. 
result  required.  -i^^T—if 

In  the  First  Solution  we  multiply  the 
denominators  6  and  4  together  for  the  denominator,  24,  of 
the  product  ;  and  the  numerators  5  and  3  together  for  the 
numerator,  15,  of  the  product.     The  Second  Solution  shows 
the  work  in  the  usual  form. 

Since  the  given  numerators  are  factors  of  the  numerator 
of  the  product,  and  the  given  denominators  are  factors  of 
its  denominator,  we  may  cancel  like  factors  from  the  nume- 
rators and  denominators  of  the  given  fractions  (see  328,  I.). 
The  product  will  then  be  in  its  lowest  terms,  as  shown  in 
the  Third  Solution. 

Ex.  2.  Multiply  51  by  3|.  solution. 

Explanation.— We  first  re-     5f  x  3^=-^f  x  V-= W-=^-^ll 
duce  the  mixed  numbers  to 
improper  fractions,  and  then  multiply  as  in  Ex.  1. 

Ex.  3.  What  is  the  product  of  |  x  4|  x  8  ? 

Explanation.  —  We  reduce 
the  mixed  number  4^  to  an  solution. 

improper  fraction,  the  integer      |  x^x  ^=;^x|^x  f  =^^ 
8  to  the  form  of  a  fraction  by 
writing  1  for  its  denominator,  and  then  multiply  as  in  Ex.  1. 

Notes. — ^1.  The  word  of  between  fractions  signifies  multiplication.  Thus, 
f  of -,%=fXi%  or  -,%Xf ;  f  of  ll=|Xll  or  llXf. 

2.  When  a  fraction  is  connected  to  any  other  number  by  of  the  expres- 
sion is  commonly  called  a  Compound  Fraction  ;  as  |  of  4-  of  2%,  f  of  13|, 
I  off  of  18. 

PROBLEMS. 

9.  Multiply  f  by  ^,  and  f  by  if.  /^,  ^f . 

10.  How  much  is  y^  of  -f  f  of  a  mile  ?  '  f  f  mi. 

11.  Multiply  7f  by  4|.  3^. 

12.  Multiply  4^  by  Q-^-^,  ^  by  3^-^,  and  18^  by  9. 

13.  The  factors  are  |f  and  ^.     What  is  the  product  ? 

14.  What  is  the  product  of  |  of  f  of  |  ?  I. 


198  FEACTIONS. 

335.  From  333  and  334  we  deduce  the 

2iule  for  Multiplication  of  ^Fractions, 

I.  Reduce  mixed  numbers  to  improper  fractions,  and  integers 
to  the  form  of  fractions. 

II.  Multiply  all  the  numerators  together  for  the  numerator, 
and  all  the  denominators  for  the  denominator,  of  the  product. 

m  OBJLBMS. 

15.  A  fruit  dealer  put  up  30  baskets  of  peaches,  puttmg  f  of  a 
bushel  in  each  basket.  How  many  peaches  did  he  put  in  all  the 
baskets  ?  28^  lu. 

IG.  If  a  man  earns  $78  in  a  month,  how  much  will  he  earn  in  |- 
of  a  month  ?  S58^. 

17.  A  man  who  owned  f  of  a  vessel,  sold  |-  of  his  share.  What 
part  of  the  vessel  did  he  sell  ?  -^^. 

18.  How  much  will  |-  of  a  yard  of  linen  cost,  at  $|^  a  yard  ? 

19.  Multiply  Hi  by  f .  ^|-|. 

20.  How  much  is  8  times  ^  ?  7^i-. 

21.  John's  kite  string  is  118  yards  long,  and  Frank's  is  ^  as  long. 
What  is  the  length  of  Frank's  kite  string  ? 

22.  What  is  the  product  of  43  multiplied  by  -^V  ?  16^^. 

23.  How  many  days'  work  can  54  men  do  in  |  of  a  day  ? 

24.  Wliat  is  the  product  of  f  x  -|-  x  f ;  or  what  is  the  cube  of  f  ? 

25.  How  much  will  4f  bu.  of  sweet  potatoes  cost,  at  %1^  a  bushel  ? 

26.  How  many  square  rods  are  there  in  a  lot  15f  rd.  long  and 
12f  rd.  wide  ?  198^. 

27.  What  will  be  the  cost  of  ff  A.  of  land,  at  $156  an  acre  ? 

28.  What  is  the  product  of  -j^  of  f  of  ^  ?  ^. 

29.  If  it  takes  1|  bu.  of  wheat  to  seed  1  acre,  how  many  bushels 
will  it  take  to  seed  17f  acres  ?  33^, 

80.  If  in  talking,  a  man  speaks  75  words  in  a  minute,  how  many 
words  will  he  utter  in  ^^  of  a  minute  ? 

31.  Raise  ^  to  the  fourth  power,  f  to  the  sixth  power,  f  to  the 
fifth  power,  and  square  ff . 


DIVISION. 


199 


32.  f  of  ^  of  f  of  a  ream  of  paper  is  what  part  of  a  ream  ? 

33.  A  railroad  train  ran  at  the  rate  of  32  miles  an  hour,  for  5-^ 
hours.    How  far  did  it  run  ?  116§^  Tni. 

34.  If  it  takes  a  man  f  of  a  day  to  mow  an  acre  of  grass,  how 
long  will  it  take  him  to  mow  -^^  of  an  acre  ? 

35.  Cube  6^,  and  square  16f.  ^Uwt^  ^^^«- 

36.  What  is  the  product  of  f  of  f  of  |  multiplied  by  f  of  f  of  |  ? 


SECTION  V. 
S)irisTOJ\r. 


FIEST  SOLUTION. 


-3  —  2 

—  7"* 


SECOND  SOLUTION. 


•2i  —7' 


TUIED  SOLUTION. 


C^SE!    I. 
The  Divisor  an  Integer. 

836.  Ex.  Divide  f  by  3. 

Explanation. — In  the  First  Solution 
we  divide  the  numerator,  6,  of  the  divi- 
dend by  the  divisor,  3  (see  319,  II.);  and 
in  the  Second  Solution  we  multiply  the 
denominator,  7,  of  the  dividend  by  the 
divisor.  The  result  in  each  solution  is 
f ,  the  required  quotient.  Since  to  di- 
vide a  number  by  3  is  to  find  I  of  it 
(see  334),  in  the  Third  Solution  we  find 
tiplication  of  fractions  (see  335),  and  the  result  is  f ,  the 
same  as  before. 

ritOBJOEMS. 

1.  Divide  I  by  5,  and  ^  by  12.  /^^  /j. 

3.  What  is  the  quotient  of  15  divided  by  ^  (=^^  ? 

3.  If  a  family  consume  5  bar.  of  flour  in  a  year,  in  what  time 
will  they  consume  1  bar.  ? 

4.  A  dealer  in  real  estate  sold  |f  of  an  acre  of  land  in  6  equal 
building  lots.    How  much  land  did  each  lot  contain  ?        ^^  A. 

5.  If  a  carpenter  can  build  13^  rd.  of  picket  fence  in  3  days, 
how  many  rods  can  he  build  in  1  day  ?  ^f . 


4  of  #  as  in  mul- 


200 


FRACTIONS. 


FIRST  SOLTJTION. 


G3     18 


C^SJS     II. 
The  Divisor  a  Fraction. 

FIRST      METHOD. 

337.  Ex.  1.  How  many  times  is  |  contained  in  7  ? 

Explanation. — Since  the  quo- 
tient is  not  changed  by  multiply- 
ing both  dividend  and  divisor  by 
the  same  number  (see  276),  wo 
multiply  them  both  by  9,  and  thus 
obtain  63  for  a  new  dividend  and 
8  for  a  new  divisor.  Then,  63  -^ 
8  =  -^g^-  =  7|,  the  required  quo- 
tient. In  the  First  Solution  the 
numbers  are  written  as  in  divis- 
ion of  integers  and  decimals ;  but  the  common  manner  of 
writing  the  numbers  is  shown  in  the  Second  Solution. 

Ex.  2.  Divide  J  by  |. 

Explanation.  — We  first  multiply 
both  dividend  and  divisor  by  4,  the 
denominator  of  the  divisor,  and  then 
divide  the  new  dividend,  -^g^,  by  the 
new  divisor,  3,  as  in  Case  I.     Hence, 

To  divide  hy  a  fraction  consists  of 
two  operations, — a  multiplication  hy  the 
denominator,  and  a  division  hy  the  nu- 
merator. 

"^        SECOND  METHOD. 

338.  We  have  seen  (975  ^^  ^^^  ^^^ )  ^^^  when  the  divi- 
sor is  a  concrete  number,  the  dividend  must  also  be  a 
concrete  number.  We  have  also  seen  (303)  that  the 
denominator  of  a  fraction  gives  denomination  or  name 
to  the  fractional  units*  We  may  therefore  regard  any 
numerator  as  one  or  more  concrete  units.     Hence, 

When  the  divisor  is  a  fraction,  the  dividend  and  divisor- 
should  he  reduced  to  similar  fractions^  before  dividing. 


SECOND  SOLUTION. 

7x9=63,  and  fx  9=8 

63^8=-%^-=7^ 

Hence,  7 -^§=7-^-. 


FIRST    SOLUTION. 
SECOND   SOLUTION. 


■h\=H 


DIVISION.  201 

■    339.  Ex.  What  is  the  quotient  of  J  divided  by  |  ? 

Explanation.  —  In    the   First 
Solution  we  reduce   both   divi-  first  solution. 

dend  and  divisor  to  similar  frac-       §-^§-= H  -^ || ='H= l-fn 
tions  (twenty-fourths),  and  then 

divide  21  twenty-fourths  by  16  second  solution. 

twenty-fourths,  in  the  same  man-         ^-^§-=^  x  §-=f-^= 1/^ 
ner  as  we  divide  21  by  16.     The 
result,  1/5,  is  the  quotient  required. 

If  we  change  the  places  of  the  terms  of  the  divisor,  and 
multiply  the  dividend,  J,  by  |,  the  fraction  thus  formed,  we 
shall  multiply  the  same  numbers  together  as  in  the  Eirst 
Solution.     This  is  shown  in  the  Second  Solution.     That  is. 

To  divide  by  a  fraction,is  the  same  as  to  change  the  places  of 
the  terms  of  the  divisor,  and  multiply  the  dividend  by  the  frac- 
tion thus  formed. 

Note  1.— When  the  places  of  the  terms  of  a  fraction  are  changed,  as  7-,  J, 
the  fraction  is  said  to  he  inverted. 

PJt  OBTjEMS. 

6.  Divide  6  by  f .  10^, 

7.  What  is  the  quotient  of  |  divided  by  f  ?  l^-^. 

8.  How  many  times  is  2^^  contained  in  3|^  ?  (^i^=^^  and  2|-=f .) 

9.  At  $1  per  cwt.,  how  many  hundred -weight  of  feed  can  be 
bought  for  $13  ? 

10.  How  many  quarts  of  chestnuts  can  be  bought  for  $f ,  at  %-^-^ 
per  quart  ? 

11.  If  a  man  can  plow  ^  A.  of  fallow  in  a  day,  how  long  will  it 
take  him  to  plow  5|^|  A.  ? 

13.  What  is  the  quotient  of  1  divided  by  f ;  or,  what  is  the 
reciprocal  of  the  fraction  f  ?  |. 

Notes.— 3,  From  this  problem  we  see  that  the  reciprocal  of  a  fraction  is 
1  divided  by  the  fraction ;  or,  it  is  the  fraction  inverted. 
3.  Division  of  fractions  is  sometimes  expressed,  by  writing  the  dividend 

above,  and  the  divisor  below  a  horizontal  line.    Thus,  -Jj-itjstjI — 5T» 
Such  expresssions  are  often  called  G&mplex  Fractions. 

9* 


202  FRACTIONS. 

310.  The  processes  developed  in  337,  338,  339,  are  all  in- 
cluded in  the  following 

^ute  for  Diyisio7i  of  Fractions, 

I.  Reduce  mixed  numbers  to  improper  fractions,  and  integers 
to  the  form  of  fractions. 

n.  Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 

PROBLEMS. 

13.  Divide  ^  by  10,  and  ||  by  16.  /^,  /^. 

14.  How  many  yards  of  gingham  ©  |^,  can  be  bought  for  $4  ? 

15.  What  is  the  quotient  of  26  divided  by  H  ?  30. 

16.  If  13  tea-spoons  weigh  -^  of  a  pound,  how  much  does  each 
spoon  weigh  ? 

17.  Divide  I  by  {-|,  and  -^  by  ^.  ^4,  12. 

18.  At  %l  a  pound,  how  much  tea  can  be  bought  for  $fi-  ?  ±f  lb. 

19.  Divide  13f  by  25,  and  16|4-  by  9.  -<|,  iff. 

20.  A  locomotive  ran  22i-  miles  in  35  minutes.    What  was  the 
rate  per  minute  ?  /^  'mi. 

21.  Divide  7|  by  f ,  and  l^V  by  -^. 

22.  What  is  the  cost  of  a  pair  of  skates,  if  ^  of  their  cost  is  $^  ? 

23.  How  many  times  is  17  contained  in  234|  ?  13  j-. 

13  455.  -• 

24.  -^j  and  ^  equal  what  numbers  ?  ^0^  lo. 

25.  If  tV  bu.  of  salt  can  be  made  from  48  gal.  of  salt  water,  how 
much  salt  can  be  made  from  1  gal.  ? 

26.  What  is  the  quotient  of  f  divided  by  3f  ?  §. 

27.  If  yV  bu.  of  mortar  cover  1  sq.  yd.  of  wall,  how  many  square 
yards  will  5^  bu.  cover  ?  63. 

28.  Divide  llf  by  3f ,  and  16^  by  6^. 

29.  If  12^  lb.  of  rice  cost  $1^\,  how  much  will  1  lb.  cost  ?    Sf 

30.  How  many  gallons  of  oysters,  at  $lf  a  gallon,  can  be  bought 

for  $n^\  ? 

31.  If  fl  oz.  of  gold  be  obtained  from  18  cwt.  of  gold  quartz, 
what  is  the  yield  from  1  cwt.  ?  /t  ^' 


REVIEW    PKOBLEMS.  203 

33.  A  lawyer's  clerk  wrote  36  pages  in  6f  hours.    How  much 
did  he  write  in  1  hour  ?  5f  pages. 

33.  I  bought  14f  qt.  of  yinegar  for  $ff .     What  was  the  price 
per  quart  ? 

34.  At  $5^  a  bushel,  how  much  clover  seed  can  be  bought  for  $f  ? 

35.  If  8f  qt.  of  strawberries  can  be  bought  for  ff|,  what  is  the 
price  per  quart  ?  Sj^o- 

36.  If  1  rod  of  fence  require  74|-  ft.  of  boards,  how  many  rods 
will  require  1811^?^  ft.  ? 

37.  ?^5!!  =  what  number  ^  /^. 

f  01  8f  ^^ 

38.  A  plank  18|  ft.  long  and  ^  ft.  thick,  contains  2f  cu.  ft.    What 
is  its  width  ?  ^ft. 


SECTION    VI. 

OiBTIBW  T'ROSI.BMS  IJV  IPHdoLC TIOJ\rS , 

1.  If  a  ship  sails  1  mi.  in  -^^  h.,  how  far  will  she  sail  in  14  h.  ? 
3.  Add  3f  t't,  '^^h  and  6f.  28-j^y. 

3.  $900  is  y\  of  what  I  paid  for  my  house  and  lot.  How  much 
did  they  cost  me  ?  $8,815. 

4.  What  is  the  difference  between  \  and  \  ? 

5.  A  miller  paid  $3,156^^  for  1,540|-  bu.  of  wheat.  What  was 
the  price  per  bushel  ?  $1^^ 

6.  A  regiment,  when  it  was  mustered  out  of  service,  consisted  of 
305  men,  which  was  -^  of  the  original  number.  How  many  men 
belonged  to  the  regiment  at  first  ?  1,087. 

7.  Add^,3:V,and|. 

8.  A  man  having  a  lot  containing  f|  A.  of^fcmd,  sold  from  it 
■^  A.  to  one  man,  and  ^  A.  to  another.  How  much  land  had  he 
left?  -^A 

9.  How  long  must  I  rent  a  house  at  $23|-  a  month,  to  cancel  a 
debt  of  $433  ?  18 j-  mo. 

10.  If  1^  rm.  of  letter  paper  cost  $f ,  what  is  the  price  per  ream  ? 


204  .    FRA^CTIONS. 

11.  A  jeweler  melted  together  ^  oz.  of  gold,  |  oz.  of  silver,  and 
^  oz.  of  copper.     How  much  did  the  mixture  weigh  ? 

12.  From  11-^  subtract  lOyV  fff. 

13.  -i\  of  I  of  f  of  15f  =  what  number  ?  5_^^. 

14.  A  man  bought  a  cow,  paying  $20i-  down,  which  was  -^  of 
the  cost.     How  much  did  the  cow  cost  ?  ^22-^. 

15.  What  is  the  sum  of  3|-|,  |,  2^, and  If  ?  P|. 
IG.  Divide  -rV  of  2|  by  -|  of  8f .                                               /_.. 

17.  A  farmer  has  4f  mi.  of  rail  fence  on  his  farm,  f^  mi.  of 
stone  fence,  ^^^j^  mi.  of  board  fence,  and  ^  mi.  of  picket  fence.  How 
many  miles  of  fence  has  he  on  his  farm  ? 

18.  The  greater  of  two  fractions  is  f  and  the  less  is  ||.  "What  is 
the  difference  ? 

19.  At  $l/o-  ^  hundred-weight,  how  much  will  it  cost  to  trans- 
port 15  hundred-weight  from  Buffalo  to  Boston  ? 

20.  The  minuend  is  1;^,  and  the  subtrahend  is  |-|.  What  is  the 
remainder  ?  of ^tt. 

21.  I  sold  a  quantity  of  wool  for  |536f,  which  W9S  1^  times  its 
cost.     How  much  did  it  cost  me  ?  $296§: 

22.  How  much  is  -^V  of  /^  of  \l  of  3^  x  4f  ?  ^  ^.f. 

23.  Bell-metal  is  composed  of  f  copper  and  ^  tin.  How  much 
of  each  of  these  metals  is  there  in  a  church  bell  that  weighs 
1^  T.  ?      ■  Copper,  -//^  T. ;    Tin,  //^  T. 

24.  Multiply  T^  by  ^ ;  -^f  by  if ;  ^  by  ii  ;  and  ^  by  iff. 

25.  How  much  will  19^  bu.  of  apples  cost,  at  the  rate  of  %A^ 

for  llf  bu.  ?  ^7|. 

26.  How  much  will  35  men  earn  in  19|^  days,  at  $ly3_  a  day  ? 

27.  How  many  loads  of  sand  at  $f  a  load,  will  pay  for  290| 
yards  of  plastering  at  %\  a  yard  ?  QS. 

28.  How  many  yards  of  cloth  \  yd.  wide,  will  line  23^  yd.,  \\  yd. 
wide  ?  S3^. 

29.  A  seamstress  bought  a  sewing-machine  for  $56.50,  paying 
$25  down.  How  much  must  she  save  from  her  earnings  each 
month,  to  pay  for  it  in  6  months  ? 


,     SECTION  I. 
cojYyb^sb  otb^atiojvs  jjv-  tub  ^ibbb^- 

BJ\rT  CZclSSBS   OIP  JVZrM2^B^S. 

34 It  Addition,  subtraction,  mnltiplication,  and  division 
are  often  called  the  Fundamental  Rules  of  Arithmetic. 

342.  Addition  is  putting  together,  and  subtraction  is 
taking  away,  or  taking  apart  ;  multiplication  is  repeated 
addition,  and  division  repeated  subtraction  of  the  same 
number.  Hence,  addition  and  multiplication  are  the 
reverse  of  subtraction  and  division. 

343.  Co7iverse  Operations  are  those  arithmetical  pro- 
cesses which  are  the  reverse  of  each  other. 


C^SE    I. 
Converse  Operations  in  the  Fundamental  Rules. 

344.  The  sum  of  the  parts  73  and  48  is  121 ;  73  +  48^:121. 

This  sum  minus  either  part  -j  i21—JiB=73  i     ^^^^^  *^® 
other  part.     Hence, 

Addition  and  subtraction  are  converse  operations. 

345.  The  product  of  the  factors  57  and  26  is  1482  ;  57  x 
26=1482. 

This  product  divided  by  either  factor  |  ;^i^|lf  ^^57 1  equals 

the  other  factor.     Hence, 

Multiplication  and  division  are  converse  operations. 

346.  From  344,  345,  we  learn  that 

I.  Either  part  is  the  difference  between  the  sum  and  the  other 
part. 


206  CONVERSE    OPERATIONS. 

II.  TJie  miniLend  is  the  sum  of  the  subtrahend  and  re- 
mainder. 

m.  Either  factor  is  the  quotient  of  the  product  divided  by  the 

other  factor. 

rV".  ^e  dividend  is  the  product  of  the  divisor  and  quotient. 

Note  1. — ^Addition  may  be  proved  by  subtraction,  and  subtraction  by  ad- 
dition. So  also  multiplication  may  be  proved  by  division,  and  division  by 
multiplication. 

PBOBJLJEMS. 

1.  The  sum  of  two  parts  is  319.5,  and  one  of  them  is  96.875. 
What  is  the  other  ?  122.625. 

2.  The  subtrahend  is  27f ,  and  the  remainder  163^.  What  is  the 
minuend  ?  -HH- 

3.  What  number  must  I  add  to  4  rd.  7  ft.,  that  the  sum  may  be 
Imi.? 

4.  The  sum  of  three  parts  is  298,  and  tvp-o  of  the  parts  are  47.5 
and  5.95.     What  is  the  other  part  ? 

Note  2. — ^Any  one  of  the  parts  is  the  difference  between  the  sum  and  the 
Bum  of  the  other  parts.  2j^J^.55. 

5.  The  sum  of  three  parts  is  43f ,  and  two  of  them  are  17f  and  f . 
What  is  the  other  part  ? 

6.  The  divisor  is  .25,  and  the  quotient  .344.  What  is  the  divi- 
dend? 

7.  The  j)roduct  of  three  numbers  is  3402,  and  two  of  them  are 
9  and  27.     What  is  the  other  number  ? 

Note  3. — Any  factor  is  the  quotient  of  the  product  divided  by  the  pro- 
duct of  the  other  factors. 

8.  The  product  of  three  factors  is  19 J,  and  two  of  them  are  If 
and  2|.     What  is  the  other  ?  5±. 

9.  The  sum  of  two  numbers  is  1,765,  and  their  difference  is  235. 
What  is  the  greater  number  ? 

Note  4.— The  sum  of  two  numbers  plus  their  difference  equals  two  times 
the  greater  number.    See  Manual.  1000. 

10.  The  sum  of  two  numbers  is  71|^,  and  their  difference  is  16^. 
What  are  the  numbers  ?  ^^f ,  27^. 


DIFFERENT    CLASSES    OF    NUMBERS.        207 


C^SE    II. 

Multiplication   and  Division  by  Factors  of  Composite 
Numbers. 

347.  Ex,  1.  Multiply  67  by  48.  solution. 

Explanation.— Since  48  =  6  times  8,  48  -^^—^  ^  ^ 

times  67  =  6  times  8  times  67,  which  is  ^ 

3216.  

Ex.  2.  Divide  3216  by  48.  "^^f 


S216 


Explanation. — Sin^e  48  is  6  times  8,  4^^ 
of  any  number  is  J  of  ^  of  the  number.  We 
find  J  of  i  of  3216  by  dividing  first  by  8  solxition. 

and  then  by  6.     Hence,  S216  ^8 

^ule  for  Multiplying  or  Dividi7ig  by  - —  ^  - 

a  Composite  JVumber.  67 

Multiply  or  divide  successively  by  any  set  of  factors  of  (he 
number. 

PJtOJiljJEMS, 

11.  Multiply  293  by  34. 

13.  How  many  square  rods  are  there  in  a  field  41.25  rd.  long  and 
35'  rd.  wide  ?  •  144S.75. 

13.  How  much  will  4.5  bu.  of  wheat  cost,  at  $1.93f  a  bushel  ? 
(45=9  X. 5.)  $8.71875. 

14.  Divide  3134  by  73. 

15.  A  peat  company  sold  54  tons  of  peat  for  $203.50.     What 
was  the  price  per  ton  ?  $3.75. 

16.  A  farmer  sowed  38  bu.  3  pk.  of  barley  on  38  A.  of  land. 
How  much  did  he  sow  to  the  acre  ?  llu.  1  pTc.  4  qt. 

17.  A  man  cleared  13f  A.  of  woodland,  cutting  49  cords  of  wood 
to  the  acre.     How  many  cords  did  he  cut*? 

18.  If  6.4  tons  of  porcelain  clay  cost  $113,  what  is  the  cost  of  .81 
of  a  ton  ?  $14.17^. 


208 


CONVERSE    OPERATIONS. 


C^SE    III. 
Multiplication  and  Division  by  Aliquot  Parts. 

348.  An  oHiquot  ^ari  of  a  number  is  any  one  of  its 
exact  divisors.  Thus,  5  is  \  of  10,  4  in.  are  \  ft.,  6  h.  are  \ 
da.,  etc. 

The  aliquot  parts  of  any  number  may  be  found  by  divid- 
ing that  number  successively  by  2,  3,  4,  5,  6,  etc. 

349.  The  Unit  of  a7i  Miquot  ^art  is  that  number 
which  is  divided  to  obtain  the  part. 

350*  TABLE  OF  ALIQUOT  PAETS. 


100 

ITon 

ift., 

1  lb. 

Aliquot  Parts  of 

1 

10 

or 
$1.00 

1000 

of 
2000  lb. 

or 
Idoz. 

of 
16  oz. 

lyd. 

1  A. 

One  half  is 

* 

5 

50 

500 

1000 

6 

8  OZ. 

1  ft.  6  in. 

80  sq.  rd. 

One  third  is 

\ 

3* 

334- 

333i 

666f 

4 

One  fourth  is 

\ 

2* 

35 

250 

500 

3 

4  oz. 

9  in. 

40  sq.  rd. 

One  fifth  is 

^ 

2 

20 

200 

400 

32      " 

One  sixth  is 

\ 

If 

16f 

1661 

333^- 

2 

One  eighth  is 

i 

n 

IH 

125 

250 

2  oz. 

4|-in. 

20  sq.  rd. 

One  tenth  is 

TO 

1 

10 

100 

200 

16      " 

One  twelfth  is 

etc. 

^ 

8| 

83^ 

1 

60LUTI0X. 

937000  {6 
156166§- 


351.  Ex.  1.  Multiply  937  by  166|. 

Explanation. — Since  166 1  is  ^  of  1000, 
166 f  times  any  number  is  J  of  1000  times 
that  number.  We  therefore  multiply  937 
by  1000,  and  divide  the  product,  937000,  by  6. 

Ex.  2.  What  will  40  sq.  rd.   of  land  cost,   at  $275  per 
acre  ? 

Explanation. — Since  $275  is  the  price  of 
1  acre,  40  sq.  rd.  or  |  A.  will  cost  \  of  $275. 
We  therefore  divide  $275  by  4. 

Ex.  3.  Divide  2775  by  33i. 

Explanation. — Since  33 1  is  |  of  100, 
33  J  is  contained  in  any  number  3  times  as 
many  times  as  100  is  contained  in  that 
number.  We  therefore  divide  2775  by 
100,  and  multiply  the  quotient,  27.75,  by  3. 


S275  [4 


SOLUTION. 

27.75 
^ 

83.25 


DIFFERENT    CLASSES    OF    NUMBEKS.         209 


SOLUTION, 


Ex  4.  If  4  eggs  cost  $.11,  what  is  the 

price  per  dozen  ?  $.11 

Explanation. — 4  eggs  are  |  of  a  dozen,  ^ 

and  1  dozen  eggs  cost  3  times  as  much  as  $.S  S 
\  dozen.     We  therefore  multiply  $.11,  the 
price  of  3  dozen,  by  3. 

352.  These  illustrations  are  sufficient  to  establish  the 
following 

^ules  for  Mtcliiplying  a7id  ^ivldinff  by  A.liquot  ^arts, 
I.  The  multipher  an  aliquot  part. 

1.  When  the  unit  of  the  aliquot  part  is  any  power  of 
10  : — Multiply  by  the  unit,  and  divide  the  product  by  the  num- 
ber of  aliquot  parts  in  the  unit. 

2.  When  the  unit  of  the  ahquot  part  is  1  : — Divide  by  the 
number  of  aliquot  parts  in  the  unit. 

n.  The  divisor  an  aliquot  part. 

1.  When  the  unit  of  the  aliquot  part  is  any  power  of 
10  : — Divide  by  the  unit,  and  multiply  the  quotient  by  the  num- 
ber of  aliquot  parts  in  the  unit. 

2.  When  the  unit  of  the  aliquot  part  is  1 : — Multiply  by 
the  number  of  aliquot  parts  in  the  unit. 

PJtOB  LJEMS. 


19.  Multiply  364  by  li 
21.  How  much  will  12|^  bu.  of 
millet  cost,  at  $3.42  a  bushel  ? 

23.  What  is  the  product  of 
333|-  times  198  ? 

25.  How  much  will  83^  A.  of 
land  cost,  at  $92  an  acre  ? 

27.  Multiply  7.14  by  l^f. 


20.  Divide  455  by  If 

22.  If  12|  bu.  of  millet  cost 
$42.75,  what  is  the  price  per 
bushel  ? 

24.  What  is  the  quotient  of 
66000  divided  by  333^  ? 

26.  If  83^  A.  of  land  cost  $7666- 
.66 1.  what  is  the  price  per  acre  ? 

28.  Divide  119  by  16f. 
29.  How  many  bushels  of  potatoes,  at  $.33^  a  bushel,  can  be 
bought  for  $19.50  ?  58.5. 


210  CONVERSE    OPERATIONS. 

30.  At  $.06|-  a  dozen,  how  much  will  144  dozen  clothes-pins  cost? 

31.  At  $.25  a  yard,  how  much  will  37.75  yards  of  shirting  come 

32.  What  is  the  cost  of  376  bushels  of  com,  at  $1.12J  per  bushel  ? 

33.  How  much  will  625  bushels  of  potatoes  come  to,  at  $.75  a 
bushel?    ($l-$i=$f=$.75.) 

34.  How  much  will  250  lb.  of  iron  cost,  at  $65  a  ton  ?    $8J2-t. 

35.  If  it  costs  $483  to  build  66f  rd.  of  Macadamized  road,  how 
much  will  it  cost  to  build  83|-  rd.  ?  $603.75. 

36.  A  gardener  raised  23  bu.  of  strawberries  from  a  piece  of  land 
8  rd.  long  and  4  rd.  wide.    "What  was  the  yield  per  acre  ?     115  du. 


SECTION   II. 

Decimals  to  Fractions,  and  Fractions  to  Decimals. 

353.  All  decimals  may  be  written  in  two  forms,  the  deci- 
mal and  the  fractional.  Thus  7  tenths  is  .7  or  /^  ;  59  thou- 
sandths is  .059  or  j  j|o  ^  ^  ten-thousandths  is  .0003  or 
TuSuTJj  ®^^'  -^^  the  decimal  form  the  denomination  or  unit 
is  indicated  by  the  position  of  the  decimal  point,  and  in  the 
fractional  form  it  is  expressed  by  the  denominator. 

Ex.  1.  Express  .075  in  the  fractional  form. 

Explanation. — ^We  write  the  number  solution. 

without  the  decimal  point,  and  express    .075  =  j^§iy  =  /g- 
its  denomination  or  unit  by  the  known 
denominator,  1000. 

BOLTITION. 

Ex.  2.  Reduce  .0081  to  the  .008^=j^f^  =  g^f^^  =  ^g^ 
fractional  form. 


CONVERSE    REDUCTIONS.  211 

7 
8 

Explanation. — Since  |  expresses  tlie  quotient      solution. 

of  7  divided  by  8,  we  annex  decimal  ciphers  to *- 

7,  and  divide  by  8,  as  in  division  of  decimals.       •'^^^ 
(See  152.) 

354.  From  these  explanations  we  deduce  the  following 

'Rules  fo7^  the  Co?ire7^se  deductions  of  decimals  and 
J^ractions, 

I.  A  decimal  to  a  fraction. 

Write  the  given  number  of  decimal  units,  omit  the  decimal 
point,  and  express  the  denomination  or  fractional  unit  by  a 
denominator. 

II.  A  fraction  to  a  decimal. 

Annex  a  decimal  cipher  or  ciphers  to  the  numerator,  and 
divide  by  the  denominator. 

PMOBLEMS. 


1.  Reduce  .375  to  a  fraction. 

3.  What  fraction  equals  .16f  ? 

5.  Eeduce  6.75  to  a  mixed 
fractional  number. 

7.  .00004  of  a  mile = what  frac- 
tional part  of  a  mile  ? 


3.  Reduce  f  to  a  decimal. 

4.  What  decimal  equals  |^  ? 
6.  Reduce  6f  to  a  mixed  deci- 
mal number. 

8. 2&l(io  of  ^  ii^il®  =  what  deci- 
mal part  of  a  mile  ? 

9.  Reduce  -^^  T.  to  the  decimal  of  a  ton.  .01875  T. 

10.  What  fractional  part  of  a  day  =  .2f  da,  ?  t^  ^' 

11.  Reduce  .06875  to  the  fractional  form. 

13.  Reduce  74^^  ^^  ^  mixed  decimal  number.  7.075. 

13.  What  fractional  part  of  a  cord  equals  .85  cd.  ? 

14.  Reduce  -^^  to  a  decimal.  .SjJ-  or  .S8^j  or  .38J^^. 

Note. — Sometimes  the  decimal  is  interminable.  In  such  cases  a  fraction 
may  be  written  after  the  decimal  figures  ;  thus,  .3^,  .73| ;  or  the  quotient 
may  be  carried  to  any  desired  number  of  decimal  places,  and  the  sign  -f- 
placed  after  it  to  show  that  the  division  is  incomplete,  or  that  there  was  a 
remainder  after  the  last  decimal  figure  of  the  quotient  was  obtained.  Thus, 
|  =  .666+;   f  =  .428571+.     SeeManuaL 


212  CONVERSE    OPERATIONS. 

C^SE     II. 

Denominate  Decimals  to  Compound  Numbers,  and  Com- 
pound Numbers  to  Denominate  Decimals. 

355.  Ex.  1.  Reduce  .75  rd.  to  a  compound  number. 

Explanation. — ^We  reduce  the  bolittiox. 

.75  rd.  to  yards  by  multiplying  .75^  ret. 

by  5.5  ;  the  decimal  part  of  this  — '— 

result,  .125  yd.,  to  feet  by  multi-  ^^'^^ 

plying  by    3  ;   and  this   result,  

.375  ft.,  to  inches  by  multiplying  '^- ^  ^  ^  2/^' 

by  12,  as  in  reduction  of  com-  

SI 5  ft 
pound  numbers  (see  %%^).     The  *    -,  J  ' 

4  rd.  and  4.5  in.  taken  together  . 

form    the    required    compound  '^* 

number,  4  rd.  4.5  in.  .     H^^^^'  '^^  ^^-  =  ^  V^'  ^'^  ^*^- 

Ex.  2.  Reduce  2  pk.  3  qt.  1  pt.  to  the  decimal  of  a  bushel. 

Explanation.  —  We  write  o«t,.^x^^ 

BOLTJTION. 

the  denominations  in  order  1.0  pt.  [2 

in  a  column,  with  the  lowest  ^  ^   i  g 

at  the  top.     We  reduce  the  ^  . 

1   pt.  to  the  decimal  of   a  — ^—  ^ 

quart  by  dividing  by  2,  as  .609376  bu. 

in  division  of  decimals,  and  Hence,  SpTc.Sqt.  lpt.=. 609375  hi. 
annex    the    result    to    the 

quarts,  making  3.5  qt.  We  reduce  the  3.5  qt.  to  the  deci- 
mal of  a  peck  by  dividing  by  8,  and  annex  the  result  to  the 
pecks,  making  2.4375  pk.  We  then  reduce  this  result  to 
the  decimal  of  a  bushel  by  dividing  by  4,  as  in  reduction  of 
compound  numbers.     (See  225-) 

Ex.  3.  Reduce  4  yd.  4.5  in.  to  /^X^ri'^ 

the  decimal  of  a  rod.  037Sft~f3 

Explanation. — Since  there  are '  ^ - 

Oft.  in  the  compound  number,  J^.r25yd. 


H  8  5 
we  write  a  cipher  in  the  place  of 


5.5 
.75  rd. 


feet  in  the    column,   and  then  27 o 

275 
proceed  as  in  Ex.  2.  


CONVERSE    REDUCTIONS.  213 

356.  From  these  explanations  we  deduce  the  following 

^utes  for  t?ie  Co7iverse  deductions  of  denominate 
Decimals  and  Com^pound  JVumbers, 

I.  A  denominate  decimal  to  a  compound  number. 

1.  Multiply  the  decimal  by  the  number  which  it  takes  of  the 
next  lower  denomination  to  equal  one  of  the  given  denomination. 

2.  Treat  the  decimal  part  of  the  product  thus  obtained  in  the 
same  manner,  and  also  the  decimal  part  of  each  succeeding 
product,  until  there  is  no  decimal  in  it,  or  until  the  lowest  de- 
nomination is  reached. 

3.  Write  the  integral  parts  of  the  several  results  and  the  final 
result  in  order,  for  the  required  compound  number. 

II.  A  compound  number  to  a  denominate  decimal. 

1.  Wi^ite  the  denominations  of  the  compound  number  in  a 
column,  with  the  lowest  at  the  top. 

2.  Divide  the  lowest  denomination  by  the  number  which  it 
takes  of  that  denomination  to  equal  one  of  the  next  higher,  and 
annex  the  result  to  the  given  number  of  the  next  higher  denomi- 
nation. 

3.  Treat  the  result  thus  obtained,  and  each  succeeding  result, 
in  the  same  manner,  until  the  whole  has  been  reduced  to  the 
required  denornination. 


mOJBZEMS. 

15.  In  .8  lb.  Troy  there  are  how 
many  ounces  and  pennyweights  ? 

17.  Keduce  .21675  of  a  ton  to  a 
compound  number. 

19.  Reduce  .26  of  a  bushel  to  a 
compound  number, 

21.  How  many  days  and  hours  in  .75  of  the  year  1875  ? 

22.  What  part  of  a  diurnal  revolution  does  the  earth  make  in 
15  h.  50  nun.  24  sec.  ?  .66. 


16.  9  oz.  12  pwt.  are  what  part 
of  a  pound  Troy  ? 

18.  Reduce  4  cwt.  33  lb.  8  oz. 
to  the  decimal  of  a  ton. 

20.  Reduce  1  pk.  .64  pt.  to  the 
decimal  of  a  bushel. 


214  CONVERSE    OPERATIONS. 

23.  How  much  wheat  must  be  sowed  upon  .85  of  an  acre,  at  the 
rate  of  1  bushel  to  the  acre  ?  S  ph.  3  qt.  ^pt. 

24.  What  part  of  a  rod  =  3  yd.  2  ft.  3  in.  ?  .5. 

25.  Reduce  4  da.  4  h.  48  min.  to  the  decimal  of  a  week. 

26.  Reduce  .45  ofa  cord  to  a  compound  number.  8cd.ft.9.6cu.ft. 

CJ^SE    III. 

Denominate  Fractions  to  Compound  Numbers,  and  Com- 
pound Numbers  to  Denominate  Fractions. 

357»  Ex.  1.  Reduce  y\  sq.  mi.  to  a  compound  number. 

Explanation. —  BOLimoN. 

We    reduce    the      t^t" «?•  mi.  x  6 40  =  ^f f^  =232^8jA, 
j\  sq.  mi.  to  acres,      /r^-  x  ^^0  =  ^ffn.  =:116/^  sq.  rd. 
by  multiplying  by      tt  «?•  rd.x30^  =  -fi-  x  ^^  =  1 1  sq.  yd. 
640 ;  the  fraction-    Hence,  /^  sq.  mi.  =  232  A.  116  sq.  rd.  11  sq.  yd. 
al  part  of  this  re- 
sult, /j  A.,  to  square  rods,  by  multiplying  by  160  ;  and  the 
fractional  part  of  this  result,  j\  sq.  rd.,  to  square  yards,  by 
multiplying  by  30|  (=  30.25) ;  as  in  reduction  of  compound 
numbers  (see  225,  I.).     The  232  A.,  116  sq.  rd.,  and  11  sq. 
yd.,  taken  together,  form  the  required  compound  number. 

Ei.  2.  Reduce  22  h.  13  min.  20  sec.  to  the  fraction  of 
a  day. 

Explanation. —  solution. 

We    reduce    the      ^^  «^^-  -^^0  =  §4  =  ^  min. 
20    sec.    to    the      ^^'^i^^-  +  i  min.  =  13^  min.  =  -^/-  min. 
fraction  of  a  min-       '^-  '^'^^^'  ^  ^  ^  =  jVk  =  f  ^• 
ute,   by  dividing      ^^ ^'  +  I  ^^-  =22§h.=^^  h. 
by  60,  and  annex      H^h.~U  =  m  =  If  da. 
or  add  the  result  Hence,  22  h.  13  min.  20  sec.  =  §fda. 

to    the   minutes, 

making  13  J  min.  We  next  reduce  the  13  J  min.,  =  y^-  min., 
to  the  fraction  of  an  hour,  by  dividing  by  60,  and  add  the 
result  to  the  hours,  making  22 1  h.  We  then  reduce  this 
result,  22 1  h.  =  ^|^  h.,  to  the  fraction  of  a  day,  by  dividing 


CONVERSE    REDUCTIONS.  215 

by  24,  as  in  reduction  of  compound  numbers  (see  225 ^  11.). 
The  final  result,  -^f  da.,  is  the  denominate  fraction  required. 

358.  From  these  examples  we  deduce  the  following 

^utes  for  the  Converse  deductions  of  Denominate 
Infractions  and  Compound  JVumbers, 

I.  A  denominate  fraction  to  a  compound  number. 

1.  Multiply  the  fraction  by  the  number  which  it  takes  of  the 
next  lower  denomination  to  equal  one  of  the  given  denomination. 

2.  Treat  the  fractional  part  of  the  product  thus  obtained  in 
the  same  manner,  and  also  the  fractional  part  of  each  succeed- 
ing product,  until  there  is  no  fraction  in  it,  or  until  the  lowest 
denomination  is  reached. 

3.  Write  the  integral  parts  of  the  several  results  and  the  final 
result  in  order,  for  the  required  compound  number. 

n.  A  compound  number  to  a  denominate  fraction. 

1.  Divide  the  lowest  denomination  by  the  number  which  it 
takes  of  that  denomination  to  equal  one  of  the  next  higher,  ex- 
press the  result  in  a  fraction,  and  annex  it  to  'the  given  number 
of  the  next  higher  denomination. 

2.  Ti^eat  the  result  thus  obtained,  and  each  succeeding  result, 
in  the  same  manner,  until  the  whole  has  been  reduced  to  the  re- 
quired denomination. 


27.  In  %r^  how  many  cents  and 
mills  ? 

29.  Reduce  i^  of  a  ream  to  a 
compound  number. 

31.  Reduce£|-|  to  a  compound 
number. 

33.  Reduce  f  of  a  square  mile 
to  a  compound  number. 


PMOBIjEMS. 

28.  In  31  cents  2.5  mills  how 
many  dollars  ? 

30.  Reduce  10  quires  16  sheets 
to  the  fraction  of  a  ream. 

82.  Reduce  10  s.  7  d.  2  far.  to 
the  fraction  of  a  pound. 

34.  Reduce  426  A.  106  sq.  rd. 
20  sq.  yd.  1  sq.  ft.  72  sq.  in.  to  the 
fraction  of  a  square  mile. 


216  CONVERSE    OPERATIONS. 

35.  What  part  of  a  bushel  is  3  pk.  |-  pt.  ?  ^^, 

36.  A  tobacco  grower  had  \\  of  an  acre  of  tobacco,  which  yielded 
at  the  rate  of  a  ton  to  the  acre.  How  much  tobacco  w^as  in  the 
crop  ?  18  cwt.  3S-^  lb.,  or  1833 l  lb. 

37.  If  11  silver  forks  weigh  1  pound  of  silver,  how  much  will  1 
set  weigh  ?  6  oz.  10  pwt.  21J>j  gr. 

38.  What  part  of  a  hogshead  is  60  gal.  2  gi.  ? 

39.  What  part  of  a  bissextile  year  is  219  da.  14  h.  24  min.  ?   |-. 

40.  How  many  powders  of  12  grains  each  will  ^  ounce  of  qui- 
nine make  ? 


SECTION   III. 
thicb,    qu^^jv ti tt,    ajv2>    cost, 

359.  In  all  transactions  of  purchase  and  sale,  and  of 
labor  and  wages,  four  elements  are  considered,  viz..  Price, 
the  Unit  of  Price,  Quantity,  and  Cost. 

360.  ^rice  is  the  sum  paid  or  allowed  for  a  unit,  or  a 
fixed  number  of  units  of  the  commodity ;  as  one,  a  dozen, 
a  hundred. 

361.  The  U7lit  of  ^?^ice  is  the  number  of  units  of  the 
commodity  upon  which  the  price  is  based. 

362.  Quantity  is  the  number  of  units  or  parts  of  a  unit 
of  the  commodity. 

363.  Cost  is  the  whole  sum  paid  or  allowed  for  the 
entire  quantity. 

C^SE     I. 


SOLUTION. 


Price  and  Quantity  given,  to  find  Cost.  ^  <?  ^  ^ 

364.  Ex.  1.  At  $3.50  a  day,  what  sum  iTf 
can  a  mechanic  earn  in  17|  days  ? 

Explanation. — In  this  example   1   day 

is  the  unit  of  price.     In  17|  days  a  man  2625' 

can  earn  17|  times  as  much  as  he  can  in  2  J,.  50 

1    day,    or    17|    times    83.50,   which    is  ^^^ 

$62,124.  $62,125 


217 


Ex.  2.  How  mucli  will  760  strawberry  plants  cost,  at  11.75 
a  hundred  ? 

Explanation. — Since  in  this  ex- 
ample 1  hundred,  is  the  unit  of 
price,  we  reduce  the  760  to  hun- 
dreds, which  we  do  by  dividing  by 
100.  Since  1  hundred,  plants  cost 
$1.75,  7.60  hundred  plants  will  cost 
7.60  times  $1.75,  or  $13.30. 

Ex.  3.  How  much  must  I  pay 
for  1968  hop  poles,  at  $43.75 
per  thousand  ? 

Explanation. — Since  1  thou- 
sand is  the  unit  of  price,  we 
first  reduce  1968  to  thousands, 
and  then  proceed  as  in  Ex.  2. 

Ex.  4.  How  much 


SOLUTION. 

760  =  7.6  0  Imndred» 

$1.75 
7.6  0 

10500 

1225 

$18.30 

SOLUTION, 

1968-^  1000=1,968 
$Ji.3.75 
1.9  68 

35000 

26250 
39375 
Ji.375 


$86.10 


SOLUTION. 

375J^-~2000  =  3.75Ji.-^ 

1.8  7  7 
$Jf-0 

$75,080 


will  3754  pounds 
of  flax  cost,  at  $40 
per  ton?  37 5 J^-~ 2000  =  3.75 1-^2  =  1.87 7 

Explanation.— 
Since  1  ton  (=2000 
pounds)  is  the  unit 
of  price,  we  first  re- 
duce 3754  pounds  to  tons,  which  we  do  by  dividing  by  the 
factors  1000  and  2  (see  347).     We  then  multiply  $40,  the 
price  of  1  ton,  by  1.877,  the  number  of  tons,  as  in  Ex.  1. 
Hence, 

The  product  of  the  price  multiplied  by  the  number  of  units 
of  price  equals  the  cost. 

PR  OBLEMS. 

1.  How  much  will  183.76  A.  of  land  cost,  at  $56.25  per  acre  ? 
3.  A  lady  bought  |  yd.  of  velvet,  at  $4.50  a  yard.    How  much 
did  it  cost  her  ?  $2.81j.. 

10 


218 


CONVERSE    OPERATIONS. 


3.  How  much  will  385  lb.  of  beef  cost,  at  $11.50  per  hundred- 
weight ?  $U'27±. 

4.  A  builder  bought  15,650  brick,  at  $9.50  per  thousand.  What 
was  the  cost  ?  $148.67^. 

5.  At  $3.50  a  ton,  how  much  will  4,G80  lb.  of  plaster  cost  ? 

365.  A  written  statement  containing  a  list  of  goods  sold 
and  their  prices  and  cost,  with  the  names  of  buyer  and 
seller,  and  the  date  of  the  transaction,  is  a  Bill.  Finding  the 
cost  of  each  item  or  article,  is  Extending  the  Item;  and  the 
total  cost  of  the  items  is  the  Footing. 

Extend  the  items,  and  find  the  footings  in  the  following 
bills: 


6. 


^i^ 


m^a- 


V       ay 


/S    ,,     ^oAfee    d/^f^ai, ,,  . /^ 

2     ^,     T^u-Z^oa/y, J 


/     £^iOOfH, 


7. 


^ou^  ^Sames  Jparsons. 

/  JS. //cf^ 

/.SO- ./JT 

^.^0 

.c."/ 

.^cT 

.SS 

.J/ 

o/a-nted  K/aidond. 


/cf- 


/S  ^</.     ty^Ceii{^??tac    *~A-6?i/d, ,,  ./S 

Si    ,,     3loac/MA ,,       S.2S 

P     ,,      ^/eitno,    Moua/  Mit/i^, ,,        /-^-^ 

//^  «^.  &//0M,  @  ^^^/   c^^<  S"^;   Ad/<y,  "^o/Zon,  /O"^. 


PRICE,    QUANTITY;    AND    COST, 


219 


C^SE     II. 


Price  and  Cost  given,  to  find  Quantity. 


1 
112 


solution:. 

2.9  2\$.56 


109 

56 

532 
50Jf. 


56 


L  lil9i 


SOLUTION. 

$86.10 

Ji-375 


$J^8.75 
1    1.9  6  8 


U2S50 

S9375 

29750 


1000 
.1968.000 


366.  Ex.  1.  How  many  pounds 
of  wool,  at  $.56  a  pound,  can  be 

bought  for  $122.92? 

Explanation. — Since  $.56  will 
buy  1  pound,  $122.92  will  buy  as 
many  pounds  as  the  number  of 
times  $.56  are  contained  in  $122.92, 
wMcb  is  219^  times. 

Ex.  2.  How  many  bop  poles  can 
be  bought  for  $86.10,  at  $43.75  per 
thousand  ? 

Explanation. — Since  $43.75  will 
buy  1  thousand  poles,  $86.10  will 
buy  as  many  thousand  poles  as  the 
number  of  times  $43.75  are  con- 
tained in  $86.10,  which  is  1.968 
times.  1.968  thousand  =  1.968  x 
1000  =  1968. 

Ex.  3.  At  $95  a  ton,  how  many 
pounds  of  iron  can  be  bought  for 
$83.60? 

Explanation. — Since  $95  will  buy 
1  ton,  $83,60  will  buy  as  many  tons 
as  the  number  of  times  $95  are 
contained  in  $83.60,  which  is  .88  of 

1  time,  or  .88  T.  We  reduce  the  .88  T.  to  pounds,  by  mul- 
tiplying it  by  2000,  and  obtain  1760,  the  required  number 
of  pounds.     Hence, 

The  quotient  of  the  cost  divided  by  the  price  of  a  unit  equals 
the  number  of  units  of  price. 


6250 

35000 
35000 


$83.60 
760 

760 
760 


SOLUTION. 

$95 


.8  8 
2000 

1760.00 


220  CONVERSE    OPERATIONS. 

PR  OBLEMS. 

8.  At  $56.25  per  acre,  how  many  acres  of  land  can  be  bought 
for  $10336.50  ?  183.76. 

9.  How  many  gallons  of  molasses,  at  |.87|-  a  gallon,  can  be 
bought  for  $37.56i  ? 

10.  My  winter's  supply  of  coal  cost  me  $48.47,  at  $9.25  a  ton. 
How  much  coal  did  I  buy  ?  5  T.  480  lb. 

11.  A  butcher  received  $44.27^  for  385  lb.  of  beef.     How  much 
did  he  receive  per  hundred-weight  ?  $11.50. 

12.  A  brick-maker  sold  brick  at  $9.50  per  thousand,  and  received 
$148.67^.     How  many  brick  did  he  sell  ?  15,650. 

13.  How  many  pounds  of  plaster,    at   $3.50   per  ton,  can  be 
bought  for  $8.19  ?  J^,  680. 

C^SE   III, 
Quantity  and  Cost  given,  to  find  Price. 
367.  Ex.  1.   If  65  army  wagons  bolution. 

cost  $23725,  what  is  the  price  of        S2S7^S  \  65^ ^ 

one  wagon?  J^ ^$365 

J,  22 
Explanation.  —  One  wagon   will  %Q0 

cost    (j'-   as   much   as   65  wagons. 

Since   65  wagons   cost   $23725,   1 

wagon  will  cost  ^^  of  $23725,  which  

is  $365. 

Ex.  2.  If  the  transportation  of  425  lb.  of  freight  cost 
$11.22,  what  is  the  price  per  cwt.  ? 

Explanation. — Since   1  cwt.   (=  solution. 

100  lb.)  is  the  unit  of  price,  we  first  U^5lb,^  J^.2 5  civL 

reduce  the  425  lb.  to  cwt.    Since  ^'^I'^ol  ^"^^ 

the  transportation  of  4.25  cwt.  costs ~  ^  ^'^  4 

$11.22,  the  transportation  of  1  cwt.  255  0 

will  cost   as  many  dollars  as  the  

number  of  times  4.25  is  contained  7700 

in   11,22,  which  is  2.64  times,  or  

$2.64. 


325 


PRICE,    QUANTITY,    AND    COST.  221 

Ex.  3.  What  is  tlie  price  per  ton  for  liay,  when  1680  lb. 

cost  $12.18  ?  SOLUTION. 

Explanation.— Since  1  ton  (=r  IGSOlh.-^ 20  0  0=.8Jf  T. 

2000  lb.)  is  the  unit  of  price,  we  $12.18    ,8  Jf 

first  reduce  1680  lb.  to  tons.    We  ^ -^        $1J^M0 

now  have  $12.18  the  cost  of  .88  T.,  318 

and  we  find  the  cost  of  a  ton,  as  ^^^ 

in  Ex.  I.,  by  dividing  the  cost  by  Jj.  ^0 

the  quantity.     Hence,  -^^^ 

Tlie  quotient  of  the  cost  divided  by  the  quantity  expressed  in 
units  of  price,  equals  the  price. 

PMOBIjEMS. 

14.  What  price  per  acre  must  he  paid  for  183.76  acres  of  land, 
to  have  it  cost  $10336.50  ?  $56.25. 

15.  If  f  yd.  of  velvet  cost  $2.81| ,  wliat  is  the  price  per  yard  ? 

16.  At   $11.50  per  cwt.,  how  much  beef  can  be  bought  foi 
$44.27^  ?  385  lb. 

17.  If  I  pay  $148.67^  for  15650  brick,  what  is  the  price  per 
thousand  ?  $9.50. 

18.  What  is  the  price  per  ton  for  plaster,  when  4680  lb.  cost 
$8.19  ?  $3.50. 

368.  Upon  the  principles  deduced  in  364,  366,  367,  are 

based  the 

Allies  for  the  Conre^^se  OperaUo7is  in  ^rice,  Quantity, 
and  Cost. 

I.  Price  and  quantity  given,  to  find  cost. 
Beduce  the  quantity  to  units  of  price,  and  multiply  the  price 

by  this  result. 

II.  Price  and  cost  given,  to  find  quantity. 
Divide  the  cost  by  the  price  of  a  unit. 

III.  Quantity  and  cost  given,  to  find  price. 

Beduce  the  quantity  to  units  of  price,  and  divide  the  cost  by 

this  result. 

Note  1. — In  business,  the  abbreviation  C.  is  often  used  for  100,  and  M. 
for  1000.     See  Manual. 


222  CONVERSE    OPERATIONS. 


PBOBLEMS. 

19.  How  much  will  3  lb.  8  oz.  of  opium  cost,  at  $4.75  a  pound  ? 

Note  2.— Since  1  pound  is  the  unit  of  price,  we  reduce  the  ounces  to  the 
decimal  or  fraction  of  a  pound. 

20.  $16.62^  will  buy  how  many  pounds  of  opium,  at  $4.75  a 
pound  ? 

21.  An  apothecary  paid  $16.62|^  for  3  lb.  8  oz.  of  opium.     What 
was  the  price  per  pound  ? 

22.  I  bought  765  pickets  for  my  door-yard  fence,  at  $1.12|^  per 
C.    How  much  did  they  cost  me  ?  $8.60f. 

23.  At  $4.50  a  yard,  how  much  velvet  can  be  bought  for  $2.81^  ? 

24.  How  much  will  a  cigar  maker  receive  for  making  13,450 
cigars,  at  $7.50  per  M.  ?  $100.87-U 

25.  A  farmer  sold  3,575  pounds  of  hay,  at  $12.50  a  ton.     How 
much  did  he  receive  for  it  ? 

26.  A  hotel  keeper  paid  $22.34f  for  hay,at  $12.50  per  ton.  How 
many  pounds  did  he  buy  ? 

27.  A  teamster  paid  $22.34|  for  3,575  pounds  of  hay.      What 
was  the  price  per  ton  ? 

28.  A  man  dug  a  cellar  28  ft.  long,  24  ft.  wide,  and  8  ft.  deep,  at 
$.66f  a  cubic  yard.  How  much  did  the  job  amount  to  ? 

29.  When  wood  is  $3.75  per  cord,  how  much  can  be  bought  for 
$2.81^?  ^cd. 

.  30.  A  hardware  merchant  paid  $61.68|  for  27^^^  gross  of  ward- 
robe hooks.     What  was  the  price  per  gross  ?  $2.25. 

31.  A  farmer  paid  $196.42  for  10,675  black-ash  rails.     What  was 
the  price  per  C.  ?  $1.8J^. 

32.  A  potter  bought  6,720  lb.  of  porcelain  clay,  at  $18  a  ton. 
How  much  did  it  cost  him  ?  $60.^8. 

33.  A  plank-road  5  mi.  235.2  rd.  long  was  built,  at  a  cost  of 
$12473.62|.     What  was  the  cost  per  mile  ?  $2, 175. 

34.  A  merchant  paid  $250.04  for  the  gas  burned  in  his  store  in 
one  year,  at  $4.75  a  thousand  feet.     How  much  did  he  burn  ? 

35.  A  paper  manufacturer  paid  $46.25  for  1,480  pounds  of  rags. 
What  was  the  price  per  ton  ?  $62.50. 


ANALYSIS.  223 

36.  At  10  s.  4  d.  sterling  per  bushel,  how  much  will  59|  bu.  of 
wheat  cost  ?  &30  IJ^  s.  10  d. 

37.  If  it  costs  $170.10  to  stereotype  a  book  of  252  pages  of 
1,080  ems  each,  how  much  is  that  per  1,000  ems  ?  $.62^^^. 

38.  How  much  lumber,  at  $24  per  M.,  can  be  bought  for  $258  ? 

10.75  M. 


SECTION    IV. 

A.  J\r^  Z  r  S  IS  . 

369.  The  method  of  stating,  in  order,  the  reasons  for  all 
tlie  different  steps  in  the  solution  of  problems,  is  often  called 
Solving  Problems  by  Analysis.     See  Manual. 

Ex.  If  3  barrels  of  flour  cost  $34.50,  how  much  will  8 
barrels  cost  ? 

Explanation. — 1  barrel  will  cost  -|  as  much  solution. 
as  3  barrels,  and  8  barrels  wiU  cost  8  times  as    $84-50  {S 

much  as  1  barrel.    -J  of  $34.50,  the  cost  of  3  $11,50 

barrels,  is  $11.50,  the  price  of  1  barrel ;  and  8 

8  times  $11.50,  the  price  of  1  barrel,  is  $92,  $ Q200 
the  cost  of  8  barrels.     Hence, 

370i  ^ule  for  Solvmg  Problems  by  Analysis, 

I.  From  the  number  and  value  of  the  things  given,  find  the 
value  of  a  unit  of  the  thing  required. 

II.  From  this  value,  find  the  value  of  the  entire  number  of 
units  of  the  thing  required. 

ritOBLE^lS. 

1.  If  6  men  lay  21  rods  of  stone-wall  in  a  day,  how  many  rods 
can  9  men  lay  ? 

3.  If  9  men  lay  31.5  rods  of  stone-wall  in  a  day,  how  many  rods 
can  6  men  lay  ? 

3.  If  6  men  lay  21  rods  of  stone-wall  in  a  day,  how  many  men 
will  be  required  to  lay  31.5  rods  ? 


224 


CONVEKSE    OPERATIONS. 


4.  If  9  men  lay  31^  rods  of  stone-wall  in  a  day,  how  many  men 
will  be  required  to  lay  21  rods  ? 

5.  When  f  yd.  of  velvet  costs  $5,  how  much  will  |-  yd.  cost  ? 

6.  If  f  T.  of  hay  costs  $13.75,  what  will  1,745  lb.  cost  ? 

7.  If  20  men  can  do  a  piece  of  work  in  12  days,  how  many  daya 
will  it  take  15  men  to  do  3^  times  as  much  work  ? 

8.  If  2  lb.  10  oz.  of  wool  make  2^  yd.  of  cloth  1}  yd.  wide,  how 
much  wool  will  it  take  to  make  150  yd.  If  yd.  v/ide  ? 

Note. — More  practice  can  be  had,  by  solving  tlie  converse  of  each  of  the 
last  four  problems. 


SECTION  V. 

Z  OJ\rG  I  TU^D  B    ;AJ\r^     TIMB. 

371.  The  circumference  of  noon. 

any  circle  may  be  divided 
into  360  equal  parts,  called 
degrees.     (See  242). 

The  equator  of  the  earth 
maybe  divided  into  24  equal 
parts  of  15°  each  (360°  ~ 
24  =  15°),  as  shown  in  the 
cut.  Since  the  earth  re- 
volves on  its  axis  from  west 
to  east  once  in  24  h.  ( =1  da), 
the  sun  appears  to  pass  round 
the  earth  from  east  to  west 
in  the  same  time,     see  Manual. 

Since  the  sun  appears  to  pass  round  the  earth  (360°) 
in  24  h.,  it  appears  to  pass  over  15°  {j=-h  of  360)  in  1  h., 
15'  (  =  6^  of  15°)  in  1  min.,  and  15"  (  =  5^  of  15')  in  1  sec. 
Consequently,  all  places  on  the  earth  change  their  relative 
position  to  the  sun  15°  in  1  h.,  15'  in  1  min.,  and  15"  in 
1  sec.  ;  and  the  relative  position  of  any  place  to  the  sun 
determines  the  time  at  that  place. 


LONGITUDE    AND    TIME/'        "  225 

372.    TABLE    OF   LONaiTUDE   AND    TIME. 

15°  difference  in  longitude  makes  1  h.  difference  in  time. 
15'  '^  "  "  "  Imin.       "        "       " 

15"         "  "  "  "  1  sec.         "        "       " 

373.  Ex.  1.  The  difference  in  time  between  "Washington 
and  London  is  5  h.  7  min.  46  sec.  What  is  the  difference 
in  longitude. 

Explanation. — Since  every  second  solution. 

of  difference  in   time   makes   15"  5  h.    7  min.  4. 6  sea 

of  difference  in  longitude  ;   every IS 

minute  of  difference  in  time,  15'  of  76°  5  6'  SO" 
difference  in  longitude ;  and  every  hour  of  difference 
in  time,  15°  of  difference  in  longitude  ;  and  since  either 
factor  may  be  used  as  the  multiplier,  we  multiply  5  h.  7  min. 
46  sec.  by  15.  The  result,  76°  56'  30",  is  the  required  differ- 
ence in  longitude. 

Ex.  2.  The  difference  in  longitude  between  Washington 
and  London  is  76°  56'  30".     What  is  the  differ-,  e  in  time  ? 

Explanation. — Since   every  solution. 

15°  of  difference  in  longitude         76°     66 '  SO"  \15 

makes   1   h.   of  difference   in  5  /j.       7  rf^^in.  ^  6  sec. 

time;  every  15'  of  difference  in 

longitude,  1  min.  of  difference  in  time ;  and  every  15"  of 
difference  in  longitude,  1  sec.  of  difference  in  time  ;  we 
divide  76°  56'  30"  by  15.  The  result,  5  h.  7  min.  46  sec, 
is  the  required  difference  in  time. 

374.  Utiles  for  the  Converse  deductions  of  difference 
in  Z/ongitude  and  Time. 

L  Difference  in  Time  to  Difference  in  Longitude. 
Multiply  the  time  by  1^;  observing  that  when  seconds,  min- 
utes, and  hours  of  time  are  multiplied,  the  respective  products 
are  seconds,  minutes,  and  degrees  of  longitude. 

II.  Difference  in  Longitude  to  Difference  in  Time. 

Divide  the  longitude  by  XS ;  observing  that  when  degrees, 
10* 


226 


CONVERSE    OPERATIONS. 


minutes,  and  seconds  of  longitude  are  divided,  the  respective 

quotients  are  hours,  minutes,  and  seconds  of  time. 

Note.— The  time  is  later  at  the  easterly,  and  earlier  at  the  westerly  of  any 
two  given  places. 


I'll  OBIjJiJM 


1.  The  difference  in  time  be- 
tween Chicago  and  New  York  is 
55  min.  44  sec.  What  is  the  dif- 
ference In  longitude  ? 

3.  When  it  is  12  o'clock  M.  at 
St.  Louis,  it  is  1  h.  20  min.  24 
sec.  P.M.  at  Portland,  Me.  What 
is  the  difference  in  longitude  ? 

5.  It  is  1  h.  2  min.  52  sec. 
P.M.  at  Richmond,  Va.,  77°  27' 
W.,when  it  is  12  o'clock  M.  at 
St.  Paul,  Minn.  What  is  the 
longitude  of  St.  Paul  ? 

7.  When  it  is  12  o'clock  M.  at  the  Island  of  St.  Helena,  5°  54'  ^Y. 
longitude,  what  is  the  time  at  Washington,  77°  3'  30"  W.  longitude? 

8.  The  time  at  Quito,  78°  50'  W.,  is  1  o'clock  P.  M.,  when  it  is 
10  h.  7  min.  20  sec.  A,  M.  at  Sacramento  City.  What  is  the  longi- 
tude of  Sacramento  City  ?  122°  W. 


2.  The  difference  in  longitude 
between  Chicago  and  New  York 
is  13°  56'.  What  is  the  differ- 
ence in  time  ? 

4.  St.  Louis  is  90°  25'  west 
longitude,  and  Portland  is  70° 
19'  west.  What  is  the  difference 
in  time  ? 

C.  When  it  is  12  o'clock  M.  at 
St.  Paul,  93°  10'  W.,  what  is  the 
time  at  Richmond,  77°  27'  W.  ? 


SECTION    VI. 
O^^E^A.  TJOJVS. 


1.  The  product  is  55f,  and  the  multiplier  4f .     What  is  the  mul- 
tiplicand ?  i^|_, 

2.  A  farmer  sheared  259  lb.  of  wool  from  56  sheep.     What  was 
the  average  weight  of  the  fleeces  ? 

3.  How  much  carpeting  will  be  required  for  a  flight  of  stairs  of 
17  steps,  each  10  in.  wide  and  8  in.  high  ?  8i  yd. 


REVIEW    PROBLEMS.  227 

4.  Reduce  .06875  to  the  fractional  form.  -4-J-^, 

5.  Eeduce  18f  sq.  rd.  to  tlie  decimal  of  an  acre.  .115  A. 

C.  How  many  pounds  of  potash,  at  $85  a  ton,  can  be  bought  for 
$37.18|  ?  815. 

7.  How  many  inches  are  there  in  .00|^  of  a  mile  ?  211.2. 

8.  A  farmer  raised  23  bu.  2  pk.  5  qt.  of  clover  seed,  and  sold  it 
at  $6.50  a  bushel.     How  much  did  he  receive  for  it  ?       $153.77. 

9.  Four  men  paid  $575  for  a  thrashing-machine.  A  paid  $175, 
]B,  $125,  and  C,  $140.     How  much  did  D  pay  ? 

10.  It  cost  $4,812  to  dig  a  sewer  If  mi.  long,  6  ft.  wide,  and 
10  ft.  deep.     What  was  the  price  per  cubic  yard  ?  $.21. 

11.  A  butcher  bought  three  beeves  on  foot,  weighing  1,463  lb., 
1,521  lb.,  and  1,584  lb.,  at  $5.75  per  cwt.  How  much  did  they  cost 
him? 

12.  I  paid  an  ice  dealer  $8.10  for  supplying  me  with  15  lb.  of 
ice  daily,  Sundays  excepted,  tor  24  weeks.  What  was  the  price 
perC?  #.^7|. 

13.  The  product  of  five  factors  is  13,  and  four  of  them  are  4f, 
1.25,  ^,  and  2.     What  is  the  other  factor  ? 

14.  A  blacksmith  paid  $170.45  for  3,896  pounds  of  bar-iron. 
What  was  the  price  per  ton  ?  $87.50. 

15.  When  it  is  20  min.  past  3  o'clock  P.  M.  at  Albany,  N.  Y., 
73°  42'  W.,  what  is  the  time  at  Athens,  Greece,  23°  44'  E.  ? 

16.  A  rectangular-shaped  farm  of  72.4  acres,  is  90.5  rods  wide. 
What  is  its  length  ?  128  rd. 

17.  What  is  the  rate  of  speed  of  a  railroad  train  that  runs  117 
mi.  in  5  h.  12  min,  ?  22.5  mi.  per  Tiour. 

18.  A  dealer  bought  417  T.  of  coal  by  the  long  ton,  at  $4.65  a 
ton,  and  sold  it  at  $5.75  per  short  ton.  How  much  did  he  gain  by 
the  transaction  ?  $746.4S. 

19.  A  cabinet  maker  paid  $112.50  for  cherry  lumber,  at  $60  per 
M.     How  much  did  he  buy  ?  1, 875  ft. 

20.  Reduce  -^^  to  a  decimal. 

21.  From  Dayton,  Ohio,  due  south  to  St.  Marks,  Fla.,  is  .024^  of 
the  earth's  circumference.     How  many  miles  is  it  ?  601.692. 

22.  Reduce  17  cwt.  44  lb.  11  oz.  to  the  fraction  of  a  ton. 


228  CONVERSE    OPERATIONS. 

23.  A  gardener  bought  45  bushels  of  potatoes  when  they  were 
worth  $.56^  a  bushel,  agreeing  to  pay  in  kind,  bushel  for  bushel, 
the  next  year.  At  the  time  of  making  payment,  potatoes  were 
worth  1.87^  a  bushel.     How  much  did  he  lose  by  the  transaction  ? 

24.  My  parlor  is  13  ft.  x  19  ft.  6  in.,  and  I  wish  to  carpet  it  with 
Brussels  carpeting,  which  is  26  inches  wide.  How  much  will  my 
carpet  cost,  at  $1.87|^  a  yard,  running  measure  ? 

25.  How  much  gold  can  be  obtained  from  a  ton  of  quartz  rock, 
if  it  yields  ^^  of  its  weight  in  gold  ?         3  lb.  10  oz.  18  imt.  8  gr. 

26.  My^xw^t  ^//a^  /p,  /cf^ 

^S2S    ,,        Moo/  ^oaic/j,  ,,  /cf 


27.  A  merchant  leaves  New  Orleans,  89°  45'  W.,  for  Augusta, 
Ga.,  81°  51'  W.  Does  he  find  his  watch  too  slow,  or  too  fast,  on 
arriving  at  Augusta,  and  how  much  ? 

28.  A  dry-goods  merchant  bought  9  pieces  of  French  calico, 
averaging  36  yd.  each,  at  $.16f  a  yard.  How  much  did  his  purchase 
amount  to  ?  $^% 

29.  What  is  the  value  of  a  hide  that  weighs  112  lb.,  at  $.08^  per 
pound  ? 

30.  A  farmer  drew  five  loads  of  hay  to  market.  The  loads  with 
the  wagon  weighed,  respectively,  3,1801b.,  3,3141b.,  3,0971b.,  2,967 
lb.,  and  3,234  lb.,  and  the  wagon  weighed  1,142  lb.  How  much  did 
the  hay  amount  to,  at  $16.50  a  ton  ?  $83.18. 

31.  A  miner  obtained  $85.78  in  silver,  from  a  quartz  rock,  the 
yield  being  at  the  rate  of  $127.65  per  ton.  What  was  the  weight 
of  the  rock?  13kklb. 

32.  What  must  be  the  width  of  a  bin  9  ft.  long  and  ^  ft.  high, 
to  contain  W  times  as  much  as  a  bin  8  x  5  x  6  ft  ?  '^jjfi- 


SECTION   I. 

375.  The  term  ^er  Cent  in  business  transactions  sig- 
nifies hundredths  of  any  thing  or  number.  Thus,  17  per 
cent  is  17  hundredths  or  17  of  every  100,  29  per  cent  is  29 
hundredths,  66f  per  cent  is  ^^i  hundredths,  etc. 

376.  Per  cent  may  be  appHed  to  any  number,  great  or 
small,  concrete  or  abstract.     Thus, 

40  per  cent  of  1  bushel  =.40  bu. ; 
88     "         "     27  miles    =.88  of  27  mi. ; 
14f  "         "     395  days  =.14|  of  395  da. ; 
7     "         "     $85.42^     =.07  of  $85.42; 
65     "         "     931-  =.65  of  93i 

377.  !Eate,  or  !Eate  "Pe?^  Ce7ity  is  the  number  which 
expresses  the  per  cent  or  number  of  hundredths. 

378.  The  term  ^erce7itage  has  two  significations  : 
1st.  It  is  the  process  of  finding  any  per  cent  of  a  num- 
ber ;  and 

2d.  It  is  the  name  of  the  result  of  the  computation. 

379.  The  ^ase  is  the  number  on  which  the  percentage 
is  computed. 

380i  The  cimouni:  is  the  base  plus  the  percentage. 
381.  The  difference  is  the  base  minus  the  percentage. 

Example. — 24  per  cent  of  50  cords  of  wood  is  .24  of 
50  cords,  or  12  cords. — ^In  this  example,  24  per  cent  is  the 
rate  ;  50  cords,  the  6ase  ;  12  cords,  the  percentage  ;  62  cords 
(=50  +  12),  the  amount;  and  38  cords  (=  50  —  12),  the 
difference. 


230 


PERCENTAGE. 


382.  Tlie  Commercial  Si(/n,  %,  when  written  after  a 
number,  signifies  per  cent. 

383.  In  computations,  any  per  cent  less  than  100  is 
expressed  by  a  decimal  or  a  fraction  ;  and  100  per  cent  or 
more,  by  an  integer,  a  mixed  number,  or  an  improper  frac- 
tion.    Thus, 


15  per 
6 

161 

i 
100 
300 
125 

233A- 


cent  or 


15^  is  ex 
H 


dOOfo 

125fo 
233|^ 


:pressed  .15  or  -^-^  =  -^ ; 
.OGor^f^; 
.045,.04|or^; 
.16forAo^  =  i; 
.0025,  .OOior^; 
oriU; 


orft^; 


1 

3 

1.25   or  iff  = 

2.33^or|^  = 


Hence,  to  express  per  cen^  decimally 
I.  Two  decimal  figures  are  always  required. 

IT.  Parts  of  1  per  cent  require  decimal  figures  or  fractions 
ai  the  right  of  hundredths. 

III.  100  per  cent  or  more  requires  an  integer  or  a  mixed 
decimal  number. 

EXEMCISES. 

1.  Read  6^,  17,^,  39,^,  112;:^. 

2.  Read  21^,  12,^^,  ^fo,  ^fc 

3.  Read  7|^  31f^,  |^,  |^,  ^fc. 

4.  Write  in  both  forms  7fc,  19^,  84^,  48^,  and  92^. 

5.  Express  22^,  56^^,  2^,  5f ^,  and  lOf^  in  both  the  decimal  and 
the  fractional  form. 

6.  Write  in  both  decimal  and  fractional  forms  36f  per  cent, 
125  per  cent,  f  per  cent,  1^^  per  cent,  and  312|^  j^er  cent. 

7.  Express  decimally  the  amomit  and  the  difference  of  1,  at  6^, 
and  at  7^. 

8.  Write  the  amount  and  the  difference  of  1,  at  25^,  in  both  the 
decimal  and  the  fractional  form. 


THE    FIVE    GENEKAL    CASES.  231 


SECTION  II. 

C^SE     I. 
Base  and  Rate  given,  to  find  Percentage. 
384.  Ex.  How  mucli  is  25^  ^^»st  solution. 

of  256?  ^^^ 

.25 
Explanation. — Since  25^  of  iOQn 

any  number  is  .25,  or  |,  of  the  ^  ^  ^ 

number,  we  find  25^  of  256  by  r  i  no 

multiplying  it  by  .25,  as  sliown 

in  the  First  Solution ;  or,  by  second  solution. 

multiplying  it  by  |,  as  shown    256  x{-  =  ^  of  266  =  64 
in  the  Second  Solution.     Hence, 

The  percentage  is  the  product  of  the  base  and  rate, 

mOBLEMS. 

1.  How  much  is  20^  of  960  bushels  of  corn  ?  192  hi. 

2.  Find  12^^  of  2,548  feet  of  lumber.  318.5  ft. 

3.  What  is  33^-^  of  12,837  ? 

4.  The  silver  used  in  coinage  contains  10^^  of  alloy.  How  much 
alloy  is  there  in  7.5  ounces  of  silver  coin  ?  .75  oz. 

5.  A  builder  bought  8  boxes  of  glass,  each  containing  45  panes ; 
but  upon  opening  them,  he  found  7|^^  of  the  glass  broken.  How 
many  panes  were  broken  ? 

6.  A  farmer  harvested  540  bushels  of  oats  from  one  field,  and 
105^  of  that  amount  from  another.  How  many  bushels  did  he 
harvest  from  the  second  field  ?  567. 

7.  From  a  hogshead  that  contained  125  gallons  of  molasses,  a 
grocer  lost  2^^  by  l,eakage.     How  much  molasses  did  he  lose  ? 

8.  What  is  f  ^  of  5,000  cords  of  wood  ?  J^5A  cd. 

9.  Flaxseed  contains  11^  of  oil.  How  much  linseed-oil  is  there 
in  275  pounds  of  flaxseed  ? 


232  PERCENTAGE. 

C^S  JE     II. 
Base  and  Percentage  given,  to  find  Rate. 

385.  Ex.  The  base  is  275,  and  the  percentage  is  6Q.  What 
is  the  rate  ? 

Explanation.— The   percent-  ^^^^  solution. 

age  on  1  is  24  5  of  the  percent-  ^^-^^    ^H 

age  on  275.     Since  m  is  the  "i^-  [^24  =  2 4% 

•  1100 

percentage  on  275,  we  divide  i  i no 

it  by  275,  and  obtain  .24,  the 

percentage  on  1,  or  the  rate,  ^^^^^^  ^^^^^^^ 

as  shown  in  the  First  Solution.        oj)^  —  j^  =  .2^  =  24-% 
Or,  2^5  of  66  is  ^%%  ;  and  re- 
ducing this  fraction  to  a  decimal,  we  obtain  .24,  the  required 
rate,  as  shown  in  the  Second  Solution.     Hence, 

The  rate  is  the  quotient  of  the  percentage  divided  by  the  base. 

rii  O  BZJSMS. 

10.  What  fc  of  5,000  bushels  are  50  bushels  ?  1^. 

11.  17  is  what  ,^  of  51  ? 

12.  Wliat  fc  of  5,725  is  2,200  ?  40fc. 

13.  My  income  last  year  was  $1,500,  and  my  expenses  were  $1,275. 
What  ^  of  my  income  did  I  expend  ? 

14.  Of  8,900  soldiers  who  went  into  battle,  1,157  were  either  killed 
or  wounded.     What  ^  of  the  army  was  lost  ?  13^. 

15.  If  2,500  pounds  of  bell-metal  are  used  to  make  a  bell  that 
weighs  2,450  pounds,  wbat  ^  of  the  bell-metal  is  waste  ? 

16.  A  grocer  sells  tea  that  cost  him  $1.20  a  pound,  ®  $1.50.    At 
what  fo  of  the  cost  does  he  sell  it  ?  125^. 

17.  What  ^  of  1  oz.  Troy  is  1  oz.  avoirdupois  ?  ^^/s-^- 

C^SE    III. 
Rate  and  Percentage  given,  to  find  Base. 

386.  Ex.  119  is  35  %  of  what  number?  solution. 
Explanation. — d5%  of  any  number  is         _?  ^  5' 

.35  of  the  number.     Since  119,  the  per-         ~TTn 
centage  of  some  number,  is  .35  times  ^t^ 

the  number — or  .35  of  the  number — we  7 


340 


THE    FIVE    GENERAL    CASES.  233 

divide  119  by  .35,  and  obtain  340,  the  required  number  or 
base.     Hence, 

The  hose  is  the  quotient  of  the  percentage  divided  by  the  rate. 

BKOB  LEMS. 

18.  465  miles  are  lo'fo  of  how  many  miles  ?  3,100. 

19.  33.13  days  are  ^fo  of  what  number  of  days  ?  365. 

30.  My  orchard  of  7.5  acres  is  6^  of  my  whole  farm.    How  much 
land  is  there  in  my  farm  ? 

31.  The  350  girls  in  a  certain  village  school  are  56^  of  the  whole 
number  of  pupils.     How  many  pupils  in  the  school  ?  625. 

33.  34  is  ^fo  of  what  number  ?  3,600. 

33.  A  shoemaker  lost  39^  of  his  property  by  a  fire,  and  his  loss 
was  $936.    How  much  was  he  worth  before  the  fire  ? 

34.  William  is  16  years  old,  and  37|^^  of  William's  age  is  40^  of 
Richard's  age.    How  old  is  Richard  ?  15  years. 

Base  and  Rate  given,  to 'find  either  Amount  or  Difference. 

387.  Ex.  If  the  base  is  375,  and  the  rate  32^,  what  is  the 
amount  ?     What  is  the  difference  ? 

SOLUTION   1. 

Explanation. — The  amount  or  the  difference  ^75 

of  375  at  any  rate  per  cent,  is  375  times  the  1.3  2 

amount  or  the  difference  of  1  at  the  same  rate.  750 

Since  the  amount  of  1  at  32^  is  1  +  .32  =  1.32,  112  5 

the  amount  of  375  at  the  same  rate  is  375  x  3  7  5 

1.32,  or  495,  as  shown  in  Solution  1.     And  ^  9  5^0  0 

Since  the  difference  of  1  at  32^^  is  1-.32  = 

.68,  the  difference  of  375  at  the  same  rate  is  solution  2. 
375 X. 68,  or  255,   as    shown  in   Solution  2.  '^^^ 

Hence,  — — 

I.  The  amount  is  the  product  of  the  base  mul-  o  o  r  n 
tiplied  by  1  plus  the  rate  ;  and 

II.  Tlie  difference  is  the  product  of  the  base       '^  ^  ^• 
multiplied  by  1  minus  the  rate. 


234  PERCENTAGE. 

I^JIOBLEMS. 

25.  If  the  base  is  125,  and  the  rate  25^,  what  is  the  difference  ? 

26.  The  base  is  63,  and  the  rate  ^%.    What  is  the  amount? 

27.  An  army  of  5,800  men  was  re-inforced  by  a  detachment  of 
39^  of  that  number.     How  many  were  then  in  the  army  ? 

28.  My  farm  contains  118.9  A.,  and  45^  of  it  is  woodland.  How 
many  acres  of  it  are  cleared  land  ?  65.395. 

29.  A  farmer  raised  625  bushels  of  wheat  one  year,  and  88^  of 
the  same  quantity  the  next  year.  How  much  wheat  did  he  raise  in 
the  two  years  ?  l^  175  lu. 

30.  I  paid  $2,400  for  a  house,  and  0^  of  that  sum  for  repairs  upon 
it.     How  much  did  the  house  cost  me  ?  $2,5^^. 

31.  A  mechanic  who  had  $147  deposited  in  a  savings-bank,  drew 
out  75^  of  it.    How  much  remained  on  deposit  ? 

32.  Last  year  the  circulation  of  a  weekly  newspaper  was  15%  less 
than  it  is  this  year,  and  this  year  its  circulation  is  14,260  copies. 
How  large  was  its  circulation  last  year  ?  12 ,  121  copies. 

Amount  or  Difference  and  Rate  given,  to  find  Base. 

388.  Ex.  1.  The  amount  of  a  certain  base,  at  18^,  is  508.58. 
"What  is  the  base  ? 

Explanation. — Any    given    amount    at  solution. 

any  rate  per  cent,  is  as  many  times  the      5  0  8,5  8     1,1 8 
amount  of  1  at  the  same  rate,  as  the  num-      U^  ^  j^g  ^ 

ber  of  times  1  plus  the  rate  is  contained  86 5 
in  the  amount.  We  therefore  divide  85^ 
508.58,  the  given  amount,  by   1.18,  the  118 

amount  of  1  at  18^,  and  obtain  431,  the  1 18 

required  base. 

Ex.  2.  The  difference  is  64.4,  ^nd  the  solution. 

rate  is  Vl\%,    What  is  the  base  ?  GU400\,8'75 

6  1  ^  5     \  

Explanation. — Any  given   difference  at     1 (^  7  3,6 

any  rate  per  cent,  is  as  many  times  the       315  0 
difference  of  1  at  the  same  rate,  as  the       -^  t>  ^  o> 


number  of  times  1  minus  the  rate  is  con-  5  2  5^0 

tained  in  the  difference.      We   therefore  ^  ^^^ 


THE    FIVE    GENERAL    CASES.  235 

divide  64.4,  the  given  difference,  by  .875,  the  difference  of 
1  at  12A^,  and  obtain  73.6,  the  required  base. 

From  these  examples  we  learn  that 

TJie  base  equals  the  quotient  of  the  amount  divided  by  1  plus 
the  rate^  or  the  quotient  of  the  difference  divided  hy  1  minus  the 
rate. 

l*It  OB  J.  JEMS. 

33.  What  number  increased  by  '7%  of  itself  is  equal  to  267.5  ? 

34.  A  horse-dealer  sold  a  span  of  matched  horses  for  $1,155, 
which  was  16  fo  less  than  they  cost  him.    What  did  they  cost  him  ? 

35.  This  year  a  clergyman's  salary  is  $2,500,  wbich  is  25^  more 
than  it  was  last  year.     What  salary  did  he  receive  last  year  ? 

36.  The  difference  is  8,466,  and  the  rate  is  15^.  What  is  the 
base?  9,960. 

37.  In  Dec.  a  manufacturer  made  4,865  yards  of  cassimere,  which 
was  12|j^  more  than  he  made  in  Nov.  How  much  did  he  make  in 
Nov.? 

38.  This  year  a  man's  house  rent  is  $325,  which  is  18|.^  less  than 
it  was  last  year.    What  rent  did  he  pay  last  year  ?  $490. 

39.  A  house  painter  painted  three  houses,  using  23|^  pounds  of 
white  lead  for  the  first  house,  which  was  20^^  less  than  he  used  for 
the  second,  and  17|^  more  than  for  the  third.  How  much  white 
lead  did  he  use  for  the  second  house  ?    How  much  for  the  third  ? 

29  lb.  6  oz. ;  20  lb. 

389.  Upon  the  principles  deduced  in  384-388  are  based 
the 

^ules  for  Co7nputations  i7i   ^erceiiiage, 

I.  Base  and  rate  given,  to  find  percentage. 
Multiply  the  base  by  the  rate. 

n.  Base  and  percentage  given,  to  find  rate. 
Divide  the  percentage  by  the  base. 

m.  Rate  and  percentage  given,  to  find  base. 
Divide  the  percentage  by  the  rate. 


236  PERCENTAGE. 

IV.  Base  and  rate  given,  to  find  either  amount  or  dif- 
ference. 

Multiply  the  base  by  1  plus  the  rate,  for  the  amount ;  and  by 
1  minus  tJie  rate,  for  the  difference, 

V.  Amount  or  difference  and  rate  given,  to  find  base. 
Divide  the  amount  by  1  plus  the  rate  ;  and  the  difference  by  1 
minus  the  rate. 

Note. — Rules  II.  and  III,  are  the  converse  of  Rule  I.,  and  IV.  and  V.  are 
the  converse  of  each  other. 

PM  OJiI.E3IS, 

40.  If  wheat  yields  72^  of  its  weight  in  flour,  how  much  flour 
can  be  made  from  245  bushels  of  wheat  ?  5^  IM. 

41.  After  drawing  9  gallons  from  a  cask  of  oil,  the  amount 
drawn  was  40^  of  the  amount  remaimng  in  the  cask.  How  many- 
gallons  were  in  the  cask  at  first  ?  3l4y. 

42.  I  paid  a  tax  of  $61.40  on  my  farm,  and  with  it  a  collector's 
fee  of  5^.    What  was  the  whole  amount  paid  2 

43.  What  %  of  423  is  75^  ?  17^fc. 

44.  If  Indian  com  contains  73^  of  starch,  how  much  starch  is 
there  in  1,192  pounds  of  com  ? 

45.  If  the  ashes  obtained  from  burning  2,275  j)ouncls  of  coal, 
weigh  68|^  pounds,  what  ^  of  the  coal  remains  in  the  ashes  ?    3fc. 

46.  A  merchant  sold  51  yards  from  a  roll  of  carpeting,  and  the 
amount  sold  was  37^^  of  the  whole  number  of  yards  in  the  roll. 
How  many  yards  were  in  the  roll  ?  136. 

47.  A  miller  bought  2,175  bushels  of  wheat,  76;^  of  which  was 
winter  wheat.     How  much  of  it  was  spring  wheat  ? 

48.  63  is  64^  of  what  number  ?  98/^75. 

49.  The  number  of  children  of  school  age  in  a  certain'county  is 
11,275,  and  3,157  children  attend  school.  AVhat  </o  of  the  whole 
number  attend  school  ? 

50.  A  merchant's  sales  for  the  year  were  1124'^  of  his  sales  for 
January,  and  his  sales  in  January  were  $1,256.  How  much  were 
his  sales  for  the  year  ?  $14117.44, 


TflE    FIVE    GENEKAL    CASES.  237 

51.  I  bought  a  house  and  lot  for  $2,750,  paying  $935  down,  and 
the  balance  in  6  equal  annual  payments.  What  <fo  of  the  purchase 
price  did  I  pay  down,  and  what  ^  at  each  annual  payment  ? 

Shi;  IH- 

52.  The  length  of  the  shadow  cast  by  a  tree  is  32^  greater  than 
the  height  of  the  tree,  and  the  tree  is  45  feet  high.  How  long  is 
the  shadow  ? 

53.  In  a  battle,  256  soldiers  were  killed.  The  number  killed  was 
20^  of  the  number  wounded,  and  the  number  wounded  was  16^ 
of  the  number  uninjured.  How  many  men  were  in  the  army  be- 
fore the  battle  ?  9,536. 

54.  Find  ^^  of  15  miles. 

55.  The  difference  between  24^  and  55;^  of  a  number  is  60.45. 
What  is  the  number  ?  195. 

56.  A  grocer  bought  a  hogshead  that  contained  110|^  gallons  of 
N".  O.  molasses.  ^^^^  of  it  leaked  out,  and  he  sold  28^  of  the  re- 
mainder.    How  many  gallons  had  he  left  ?  7<?|-. 

57.  25^  of  40^  of  a  number  is  what  part  of  the  number  ? 

lOicor-^^ofit. 

58.  A  fanner  sold  28<^  of  his  land,  and  afterward  bought  35^^  of 
as  much  as  he  had  left.  He  then  had  5|^  acres  less  than  at  first. 
How  many  acres  had  he  at  first  ?  1S74^. 

59.  What  <fo  of  27|-  is  4|  ? 

60.  One  year  a  farmer  raised  560  bushels  of  wheat,  and  sold  it 
at  $1.80  a  bushel.  The  next  year  he  raised  25^  less,  and  sold  it  at 
25^  more  per  bushel.  In  which  year  did  he  realize  the  greater 
sum  for  his  wheat  ?  Thejirst  year,  $63  more. 

61.  Of  a  regiment  of  soldiers,  4^  deserted,  and  6^^  of  the  remain- 
der were  killed.  Of  those  then  left,  16f  ^  were  taken  prisoners,  and 
12^  of  the  balance  were  discharged.  There  were  then  660  men  in 
the  regiment.     How  many  men  were  there  at  first  ?  1, 000. 

62.  A  wood  dealer  contracted  to  deliver  8,100  cords  of  wood  at 
a  R.R  station,  in  90  working  days.  When  70^  of  the  time  had 
passed,  he  had  delivered  but  65^  of  the  wood.  How  many  cords 
must  he  deliver  each  day  for  the  balance  of  the  time,  to  fulfill  the 
contract  ?  105. 


238 


PERCENTAGE. 


63.  7,465  is  33^^  of  what  number  ?  SS,395. 

64.  15fo  of  484  is  33,^  of  what  number  ? 

65.  A  manufacturer  increased  his  capital  by  24:^  the  first  year, 
and  that  capital  by  25fo  the  second  year.  The  third  year  he  lost 
IQfo  of  his  capital,  and  he  then  had  $16,317  left.  How  much  capi- 
tal had  he  at  first  ?  $12,455.45. 


SECTION    III 

INSHSMe. 


),  Insurance  is  a  security  against  loss  or  damage 
within  a  given  time,  guarantied  to  one  party  by  another, 
for  a  specified  consideration. 

391.  J^ire- Insurance  is  a  security  against  loss  by  fire. 

392.  J)farine  Insura7ice  is  a  security  against  loss 
at  sea. 

393.  Health  and  Acczde7zt  Insurance  are  securi- 
ties against  loss  by  sickness  or  accident. 

394.  Hfe-Insura7ice  is  a  security  guarantying  to 
the  parties  interested  in  the  life  of  a  person,  a  specified 
sum  at  his  death,  if  it  occurs  within  a  specified  time. 


INSURANCE.  239 

895.  Valuation  is  the  sum  for  which  property,  hfe,  or 
iiealth  is  insured. 

396.  ^remitim  is  the  sum  paid  for  the  insurance. 

397.  The  Policy  is  the  contract  between  the  insurer 
and  the  insured. 

Notes.— 1.  The  business  of  insuring  is  commonly  carried  on  by  corpo- 
rations called  Insurance  Companies. 

2.  A  corporation  whose  members  have  paid  in  money  or  capital,  to  secure 
the  payment  of  losses,  and  among  whom  the  profits  are  divided,  is  a  Stock 
Insurance  Company. 

3.  A  corporation  of  which  every  person  insured  is  a  member,  sharing  in 
the  profits  and  losses,  is  a  Mutual  Insurance  Company, 

4.  In  order  that  owners  of  property  insured  may  not  be  tempted  to  de- 
stroy it,  property  is  never  insured  for  its  full  value. 

com:i'tjta.tioi^s  in  xnsxjraivce:. 

398.  Valuation  is  the  base  ; 
Premium  is  the  percentage  ;  and 
Rate  %  is  the  rate.   Hence, 

I.  Valuation  and  rate}  •  j  Base  and  rate  given,  to 
given  J  to  find  premium,        )         \  find  percentage. 

II.  Valuation  and  premi-)  •  j  Base  and  pei^centage  giv- 
um  given,  to  find  rate,  )          \  en,  to  find  rate. 

in.  Rate  and  premium  )  :□  j  Bcd^  and  percentage  giv- 
given,  to  find  valuation,        f         (  en,  to  find  base. 


Pit  OBIjEMS. 

1.  What  premium  must  a  merchant  pay  for  an  insurance  of 
$7,350  on  his  stock  of  goods,  at  l]-^  ?  $90.62 f 

3.  A  school-house  was  insured  for  $3,800,  at  ^^.  What  was  the 
premium  ? 

3.  A  physician  gets  a  policy  of  insurance  on  his  house  for 
$1,500,  his  household  furniture  for  $650,  and  his  library  for  $475. 
How  much  does  it  cost  him,  at  f^  ?  $19.68^. 


240  PERCENTAGE. 

4.  If  it  costs  $521.35  for  an  insurance  of  $27,800  on  a  merchant 
vessel,  for  a  trip  from  New  York  to  Havana,  what  is  the  rate  'i 

5.  It  costs  $172.50  to  insure  a  steam  planing-mill  for  $3,450. 
What  is  the  rate  ?  5^. 

6.  The  premium  paid  for  insuring  a  paper-mill  for  $15,500,  was 
$116.25.     What  was  the  rate  ? 

7.  A  shoemaker  paid  $9.37|-  for  having  his  shop  and  stock 
insured,  at  1^%.     What  amount  did  his  policy  cover  ?  $750, 

8.  If  I  pay  $20  for  having  my  house  insured,  at  |^,  what  amount 
do  I  get  it  insured  for  ?  $3,200. 

9.  At  1|^,  what  amount  must  be  covered  by  a  policy  that  costs 
$141.75  ? 

10.  A  grain  dealer  had  a  cargo  of  wheat  insured  from  Milwaukee 
to  Buffalo  for  $8,750,  at  ^fc.    What  premium  did  he  pay  ? 

11.  The  premium  paid  for  insuring  a  church  for  $12,750,  was 
$63.75.     What  was  the  rate  ?  |^. 

12.  At  1|^,  how  much  will  it  cost  to  insure  a  hotel  for  $125,000, 
and  the  furniture  for  $40,000  ? 

13.  The  premium  for  insuring  a  lake  propeller  for  the  season, 
at  1|^,  was  $31 2.37 J.     For  what  amount  was  she  insured  ? 

$17,850. 

14.  A  policy  of  $2,675  on  a  sash  and  blind  factory,  cost  $53.50. 
What  was  the  rate  ?  2%. 

15.  What  will  it  cost  per  annum  for  a  life-insurance  policy  for 
$5,000,  on  the  life  of  a  man  35  years  old,  at  $27.50  per  $1,000  ? 

Note  5. — The  Tables  of  Rates  of  most  life-insurance  companies  are  at 
certain  sums  per  $1,000  of  insurance,  tlie  rate  per  $1,000  increasing  accord- 
ing to  the  age  at  which  the  insurance  is  made. 

16.  A  man  30  years  old  has  his  life  insured  for  $3,000,  at  an 
annual  premium  of  $24.75  per  $1,000.  If  he  dies  at  the  age  of  50, 
how  much  more  do  his  heirs  receive  upon  his  life-insurance  than 
he  has  paid  on  it  ?  $1,515. 

17.  A  man  45  years  old  obtains  a  policy  of  insurance  for  $7,000, 
in  a  Mutual  Endowment  Insurance  Co.,  the  policy  to  be  paid  at 
60,  paying  at  the  rate  of  $26.84  on  every  $1,000  semi-annually. 
What  are  the  annual  payments  ?  $375.76. 


COMMISSION, 


241 


SECTION  IV. 


399.  An  Agent,  Commission- Merchant,  J^aclor, 
or  !Sroker  is  a  person  who,  by  authority,  buys  and  sells 
goods,  or  transacts  other  financial  business  for  another. 

Note. — A  person  to  whom  property  is  delivered  in  trust,  for  sale,  is  a 
Consignee  ;  and  the  person  delivering  tlie  property  is  a  Consignor. 

400.  Commission  is  the  sum  paid  an  agent  or  commis- 
sion-merchant for  transacting  business. 

COnVLFUTATIONS    IN    COMiiyniSSION". 

401.  Commission  is  commonly  computed  at  some  %  on 
the  sum  of  money  received  or  paid  out  by  the  agent  in 
the  transaction. 

402.  The  sum  on  which  commission  is  computed  is  the 
base  ; 

Commission  is  the  percentage  ; 
Rate  %  is  the  rate  ;  and 

The   sum   on  which  commission  is  computed  plus   the 
commission  is  the  amount.     Hence, 
11 


242  PERCENTAGE. 

I.  Tlie  sum  on  which  commission  )  (  Base  and  rale 
is  computed  and  the  rate  given,  toria  j  given,  to  find  per- 
find  the  commission,  J        '  centage. 

n.  The  sum  on  which  commis-  j  (  Base  and  per- 
sion  is  computed  and  the  commis-  r  is  -<  centage  given,  to 
sion  given,  to  find  the  rate,  )         {find  rate. 

ni.  Tlie  rate  and  the  commission  j  c  Bate  and  percent- 
given,  to  find  the  sum  on  which  r  is  •<  age  given,  to  find 
commission  is  computed,  )        ( base. 

rV.  Tlie  amount  and  rate  given,  )        [       a  i      j      t 

^    ,  ,  ,  .  ,  .     /   .     \      Amount  and  rate 

to  find  the  sum  on  which  commis-  }■  is  •< 

sion  is  computed,  )        I  ?»"^"' '» ■^'^  '^"'- 

P  Jt  O  B  L  EMS, 

1.  In  one  month  an  insurance  agent  receives  |1,328  for  premiums, 
and  bis  commission  is  5^.     How  much  do  his  fees  amount  to  ? 

2.  A  commission-merchant  bought  for  a  provision  dealer  420 
barrels  of  pork,  @,  $21.50,  at  l^fo  commission.  How  much  was  his 
commission  ?  $112.87  i-. 

3.  A  real  estate  agent  sold  a  farm  of  119|-  acres,  at  $96  per  acre. 
What  was  his  commission,  at  |^  ?  $71.70. 

4.  A  miller  paid  a  grain  buyer  $64.12|^,  for  buying  15,000  bushels 
of  com,  at  $.57  per  bushel.    What  rate  of  commission  did  he  pay  ? 

5.  An  auctioneer  sold  a  lot  of  crockery  for  $416,  and  received 
$18.72  commission.     What  was  his  <fo  for  selling  ?  H%' 

6.  A  merchant  paid  an  attorney  $56.70,  for  collecting  bills  to  the 
amount  of  $945.     What  <fo  were  his  fees  for  collecting  ?  6fc. 

7.  A  wool  buyer  bought  wool  at  $.44  per  pound,  and  received  a 
commission  of  $187.11,  at  l^'fc    How  much  wool  did  he  buy  ? 

8.  If  I  pay  an  agent  $192.75  for  purchasing  goods,  at  3^  commis- 
sion, what  is  the  cost  of  the  goods  purchased  ?  $6,Jj,25. 

9.  A  commission-merchant  received  $157.75  for  selling  flour, 
commission  2|^^.     How  much  did  the  flour  sell  for?  $6,310. 

10.  How  much  land,  at  $35  an  acre,  can  an  agent  buy  with 
$3126.20,  after  deducting  his  commission  of  l^^o  ?  88  A. 


PKOFIT     AND    LOSS.  243 

11.  An  agent  receives  $901.25  with  which  to  purchase  hides, 
after  deducting  his  commission  of  3^.  What  sum  will  he  invest  in 
hides  ?  $875. 

12.  A  cotton  factor  received  $4,076.80  to  be  invested  in  cotton 
at  $.28  a  pound,  after  deducting  ^fo  for  his  fees.  How  many 
pounds  did  he  buy  ?  1J^,000. 

13.  A  tax  collector  had  a  warrant  for  $37,600,  upon  which  he 
collected  $18,228,  at  1,^,  and  the  balance  at  5,^.  What  was  the 
amount  of  his  fees  ?  $1, 150.88. 

14.  A  fruit  buyer  received  $7,315  with  which  to  buy  apples, 
after  taking  out  his  commission  of  A.^fc.  How  much  did  he  use  in 
buying  apples  ?  $7,000. 

15.  A  buyer  of  live  stock  receives  $484.50  with  which  to  buy 
sheep,  after  deducting  his  commission  of  2;^.  What  sum  does  he 
expend  ? 

16.  A  produce  commission  house  in  Detroit  received  $4,725  from 
an  eastern  miller,  to  be  invested  in  wheat,  less  a  commission  of 
2|^^.     How  much  was  the  commission  ? 

17.  A  commission-merchant  who  buys  produce  at  2f;^  commis- 
sion, receives  $1350.20  with  which  to  purchase  beef.  How  nmcli 
ia  his  commission  ?  $36.  I4. 


SECTION  V. 

403.  When  goods  are  sold  for  more  than  cost,  the  excess 
is  Profit,  or  an  Advance  ;  and  when  they  are  sold  for  less 
than  cost,  the  deficiency  is  Loss,  or  a  Discount.     Hence, 

404.  "Profit y  in  business,  is  the  sum  above  cost  for 
which  goods  are  sold,  or  the  excess  of  receipts  over  expen- 
ditures ;  and 

405.  JjOSS  is  the  sum  below  cost  for  which  goods  are 
sold,  or  the  excess  of  expenditures  over  receipts. 


244  PERCENTAGE 


C0MPXJTA.T10NS    IN"    I»ROI''IT    JSJSIID    LOSS. 

406.  Profit  and  loss  are  commonly  computed  at  a  ^  on 
the  cost. 

407.  The  cost  is  the  base ; 

The  profit  or  loss  is  the  percentage  ; 

The  rate  %  is  the  rate  ;  and 

The  selling  price  is  the  amount  or  difference.     Hence, 

I.  Cost  and  rate  given,  to)  .  j  Base  and  rate  given,  to 
find  gain  or  loss,  )         (  find  percentage. 

II.  Cost,  and  gain  or  loss  )  •„  j  ^cise  and  percentage  giv- 
given,  to  find  rate,  )         (  en,  to  find  rate. 

in.  Gain  or  loss,  and)  ■  j  Percentage  and  rate  given, 
rate  given,  to  find  cost,          f         ( to  find  base. 

TV.  Cost  and  rate  given,  )  .  j  Base  and  rate  given,  to 
to  find  selling  price,              )         \  fip.d  amount  or  difference. 


V.  Selling  price  and  rate  \    .     j     Amount  or  difference,  and 
given,  to  find  cost,  )         \  rate  given,  to  find  base. 


FHOnZJSMS. 

1.  A  man  bought  a  house  and  lot  for  $1,875,  and  sold  it  at  a 
loss  of  4^.     How  much  did  he  lose  ?  $75. 

2.  If  a  butcher  buys  beef  at  $.08  per  pound,  how  must  he  sell  it 
to  gain  S7^^  ?  At  $.11  per  pound. 

3.  I  bought  a  cow  in  the  spring  for  $63.50,  and  sold  her  in  the 
fall  for  $45.     What  ^  of  the  cost  did  I  lose  ? 

4.  A  grocer  pays  $12  a  barrel  for  mackerel,  and  retails  them  at 
$.10  a  pound.     What  fo  does  he  gain  ?  ^^j%- 

5.  A  grocer  by  selling  butter  at  a  profit  of  20^,  made  $.05  on 
a  pound.     What  did  the  butter  cost  him  per  pound  ? 

6.  A  lumber  dealer  loses  $10.50  per  M.  by  selling  a  quantity  of 
lumber  at  37|-^  below  cost.  What  does  the  lumber  cost  him  per 
M.  ?  $28. 

7.  A  hardware  merchant  by  selling  a  stove  at  32^  above  cost, 
makes  $6.     What  was  the  cost  of  the  stove  ?  $18.75. 


STOCKS.  245 

8.  At  what  price  must  a  grocer  sell  cheese  that  cost  him  $.15 
per  pound,  to  gain  ddlfo  ?  $.20. 

9.  I  sold  a  watch  that  cost  me  $75,  at  a  loss  of  8^.     For  what 
price  did  I  sell  it  ?  $69. 

10.  A  carpenter  built  a  house  at  a  cost  of  $1,380,  and  sold  it  at 
a  gain  of  12|-^.    For  how  much  did  he  sell  it  ? 

11.  If  I  gain  30^  by  selling  sheep  at  $4.87|-  a  head,  how  much 
did  they  cost  me  ?  $3.75. 

12.  A  merchant  loses  12^  by  selling  damaged  delaines  at  $.33  a 
yard.     How  much  did  they  cost  him  ?  $.37-^-  a  yd. 

13.  A  builder  erected  a  church  by  contract,  for  $15,300,  and  lost 
15^  upon  its  cost.     How  much  did  it  cost  him  to  build  it  ? 

14.  How  shall  I  mark  calico  that  costs  $.1G  a  yard,  to  gain  25^  ? 

15.  A  merchant  sells  sugar  at  $.15  per  pound,  that  cost  him 
$.12|.    What  fo  does  he  gam  ?  20 fc 

16.  A  wagon  maker  sold  a  lumber  wagon  that  cost  him  $96,  at 
25fo  profit.     At  what  price  did  he  sell  it  ?  $120. 

17.  A  dealer  in  musical  instruments  sold  a  piano  for  $540,  and 
his  profit  was  20^.     How  much  did  the  piano  cost  him  ? 

18.  A  furrier  sold  a  set  of  ladies'  mink  furs  at  15^^  less  than  cost, 
and  lost  $10^  on  them.     How  much  did  he  get  for  them  ? 


SECTION  VI. 

S  TO  C^S. 

408.  A  Corpoi^ation  is  a  company  established  by  law, 
having  power  to  transact  business  as  an  individual. 

409.  Stock  is  the  property  invested  in  the  business  of  a 
corporation. 

Note.— Stock  is  often  called  Capital,  or  Capital  Stock. 

410.  A  Share  is  one  of  the  equal  parts  into  which  the 
stock  of  a  corporation  is  divided.     It  is  usually  $100. 

411.  A  Certificaie  of  Stock  states  the  number  of 
shares  of  stock  owned  by  the  holder  of  the  certificate,  and 
also  the  j)ar  value  of  a  share. 


246  TERCENTAGE. 

412.  The  ^a7*  Value  of  stock  is  the  sum  stated  in  the 
scrip  or  certificate  ;  and 

413*  The  J)fa7^ket  Tatue  is  the  sum  for  which  the 
stock  will  sell. 

414.  Stock  is  M  ^ar  when  its  market  value  is  its  par 
value,  or  100^  ; 

415.  It  is  Above  ^a7^  when  its  market  value  is  above 
its  par  value,  or  more  than  100^  ;  and 

416.  It  is  !Selow  ^ar  when  its  market  value  is  below 
its  par  value,  or  less  than  100^. 

417.  ^remiufn  is  the  excess  over  100^  in  the  value  of 
stock  that  is  above  par  ;  and 

418.  !Discou7ii:  is  the  deficiency  under  100^  in  the  value 
of  stock  that  is  below  par. 

419.  Sioc/c  QuotaHo7ls  are  published  statements  giv- 
ing the  market  value  of  stocks.  Thus,  if  stock  is  8^  above 
par,  it  is  quoted  at  108  ;  and  if  it  is  %%  below  par,  it  is 
quoted  at  92. 

420.  A  Stock  !Sroke7'  or  Stock  Jobber  is  a  person 
who  deals  in  stocks. 

421.  !S7*oke7'a(/e  is  the  commission  paid  to  stock  bro- 
kers for  buying  and  selling  stocks  for  others. 

Note. — The  rate  of  commission  established  by  the  N.  Y.  Board  of  Brokers 
is  \%  on  the  par  value  of  the  stock. 

COIMPXJT^TION^S    IN^    STOCKS. 

422.  In  stock  transactions  the  computations  are  made  on 
the  par  value  of  the  stock. 

423.  The  par  value  is  the  base  ; 

The  premium  or  the  discount  is  the  percentage  ; 

The  rate  %  is  the  rate  ;  and 

The  market  value  is  the  amount  or  difference.     Hence, 

I.  Far  value  and  rate  giv-  )         (      Base  and  rate  given,  to 
en,  to  find  premium  or  dis- ]:  is  "{  find  percentage, 
count,  ;         (      / 


STOCKS.  247 

II.  Par  value  and  rate )  .  f  Base  and  rate  given,  to 
given,  to  find  ^narket  value,  )         \  find  amount  or  difference 

III.  Market  value  and  rate)  •  j  Amount  or  difference,  and 
gwen,  to  find  par  value,        )         \  rate  given,  to  find  hase. 

These  three  cases  cover  tlie  ordinary  transactions  in 
stocks. 

PJJ  OBZEM  S. 

1.  If  I  buy  17  shares  of  bank  stock  at  par,  and  sell  it  at  5|^;^  pre- 
mium, how  much  do  I  gain  ?  $93.50. 

2.  A  man  bought  38  shares  of  the  stock  of  an  express  company 
at  par,  and  sold  it  at  life  discount.     How  much  did  he  lose  ? 

3.  How  much  will  I  receive  for  193  100-dollar  shares  of  insurance 
stock,  if  I  sell  it  at  24|;^  above  par  ? 

4.  Mr.  Clark  took  7  lOOO-doUar  shares  in  the  stock  of  a  woolen 
factory,  at  13;^  below  par.     How  much  did  it  cost  him  ?    $6,090. 

5.  If  X  exchange  65  shares  of  bank  stock  at  26^  premium,  for 
R.R.  stock  at  9^  discount,  how  many  shares  will  I  receive  ?      90. 

6.  When  State  stocks  are  quoted  at  82,  what  is  the  par  value  of 
the  stock  that  can  be  purchased  for  $2,460  ? 

7.  When  Panama  R.R.  stock  is  quoted  at  123,  how  many  shares 
can  be  bought  for  $6,642  ?  54. 

8.  A  stock  jobber  bought  50  shares  of  the  stock  of  a  coal  com- 
pany at  114|-,  and  sold  it  at  135.    How  much  did  he  gain  ? 

9.  How  many  100-dollar  Pacific  R.R.  bonds  can  be  bought  for 
$5694,  at  d^^  premium  ? 

10.  Bought  96  shares  of  the  stock  of  an  iron  mill  at  2^  discount, 
and  sold  it  at  9^^  discount.     How  much  did  I  lose  ?  $720. 

11.  A  broker  bought  76  shares  of  mining  stock  at  4:^fo  discount, 
and  sold  it  at  7^  premium.     How  much  was  his  gain  ?         $874. 

12.  A  broker  bought  84  shares  of  R.R.  stock  at  19^^  discount. 
He  sold  35  shares  at  27^^  discount,  and  the  balance  at  Sfo  discount. 
Did  he  gain  or  lose,  and  how  much  ?  $241.50. 

13.  Will  I  gain  or  lose,  if  I  buy  112  shares  of  the  stock  of  a 
transportation  company  at  17^  premium,  and  after  receiving  a  divi- 
dend of  9^^  sell  it  at  Sfc  less  than  it  cost  me  ? 


248  PERCENTAGE. 

SECTION   VII. 

T^X£^S    ^JV^    DUTIES. 

424.  Heve7iue  is  tlie  annual  income  whicli  Government 
collects  and  receives  into  the  treasury,  for  public  use. 

I.    GENERAL    TAXES. 

425.  Taxes  or  duties  are  sums  of  money  assessed 
upon  persons  and  property,  to  meet  public  expenses. 

Notes. — 1.  A  Poll-Tax  is  a  tax  upon  the  person ;  and  a  Property  Tax  is  a 
tax  upon  the  assessed  value  of  property. 

2.  Property  is  of  two  kinds  : — Real  Estate^  or  houses  and  lands ;  and  Per- 
sonal Property^  or  movable  property. 

3.  The  general  taxes  levied  or  assessed  are  Road,  School,  Village,  City, 
Town,  County,  and  State  taxes ;  and  Special  Property  Taxes  for  local  im- 
IDrovements. 

co]vii>xjT^a?iO]srs  iisj-  oener^l  taxes. 

426.  In  the  assessment  of  taxes,  assessors  must  first  find 
the  rate,  and  then  the  tax. 

427.  The  valuation  of  property  is  the  base  ; 
The  rate  %  is  the  rate  ;  and 

The  tax  is  the  percentage.     Hence, 

I.  Valuation  and  tax  giv-  )  .  j  JSase  and  percentage  giv- 
en, to  find  rate,  )         \  en,  to  find  rate. 

n.  Valuation  and  rate )  .  J  Base  and  rate  given,  to 
given,  to  find  tax,  )         I  find  percentage. 

m.  Tax  and  rate  given, )  .  j  Percentage  and  rate  given, 
to  find  valuation,  )  i  to  find  base. 

These  three  cases  cover  the  ordinary  computations  in 
general  taxes. 

PJ2  OS  JO  JEMS. 

1.  A  school  tax  of  $433.50  is  levied  in  a  district,  and  the  property 
is  assessed  at  $69,360.  What  is  the  rate  ?  .60,  or  $.006  j.  on  a  dollar. 

2.  A  tax  of  $95,935  is  levied  on  a  city,  the  assessed  valuation  of 
which  is  $7,674,800.     What  is  the  rate  ? 


TAXES    AND     DUTIES.  249 

3.  If  I  am  assessed  at  $1,250  on  a  house  and  lot,  $300  on  a  vacant 
lot,  and  $3,000  personal  property,  how  much  will  my  tax  be,  the 
rate  being  $.0097  on  a  dollar  ?  $U-1H. 

4.  The  assessed  valuation  of  a  village  is  $294,500,  and  a  tax  of 
$1,145  is  to  be  laid.     What  must  be  the  rate  ? 

5.  A  tax  of  $928.80  for  building  a  bridge,  is  levied  on  a  town, 
the  assessed  valuation  of  which  is  $967,500.  What  is  the  tax  on 
property  assessed  at  $1,250  ?  $1.20. 

6.  A  physician  whose  property  was  assessed  at  $2,750,  paid  a  school 
tax  of  $23.37|-.     What  was  the  rate  of  taxation  ? 

7.  One  year,  a  man  whose  property  was  assessed  at  $1,350,  paid 
.35^  village  tax,  .47^  school  tax,  1.05^  county  tax,  and  $1.00  poll 
tax.     What  was  the  amount  of  his  taxes  ?  $26.^4^. 

8.  If  the  rate  is  $.001^  on  a  dollar,  and  the  tax  is  $1178.85,  what 
is  the  valuation  ? 

9.  If  a  tax  of  $473.40  is  levied  on  property  assessed  at  $39,450, 
what  is  the  assessed  valuation  of  property  that  pays  a  tax  of 
$29.70?  .5-^.475. 

II.    IN-TERNAIi    REVENUE. 

428.  Internal  Hevetiue  is  the  income  whicli  Govern- 
ment receives  from  home  business,  products,  and  manu- 
factures. 

429.  Ificome  Tax  is  a  tax  levied  upon  income. 

430.  A  license  I^ee  is  a  tax  levied  for  a  license  or  per- 
mit to  carry  on  any  branch  of  business. 

431.  A  Tax  ozi  Manufactures  is  a  tax  levied  upon 
the  value  of  home  manufactures. 

COMIPXTTATIONS    IInT    INTEJE,N"AI:j    rea^enxje. 

432.  Income  taxes  are  computed  at  some  legal  rate  upon 
the  income  minus  the  exemptions  ;  and 

Taxes  on  manufactures,  at  some  legal  rate  upon  the 
value  of  the  manufactured  goods. 

License  fees  are  fixed  sums  established  by  law. 


250  PERCENTAGE. 

433.  The  assessed  income  {i.  e.  income  minus  exemptions), 
or  the  value  of  the  manufactured  goods,  is  the  base  ; 
The  rate  %  is  the  rate  ;  and 
The  tax  is  the  percentage.     Hence, 

Assessed  income,  or  value  of  ^  r  Base  and  rate  giv- 
maniifactures,  and  rate  given,  toy  is  <  en,  to  find  percent- 
find  tax,  )        ( age. 

I'M  OBLEMS. 

10.  A  lawyer's  income  for  the  year  18G8  was  $3,284,  and  his  ex- 
emptions were  $350  for  house  rent,  and  $1,000  for  living  expenses. 
How  much  income  tax  did  he  pay,  the  rate  being  5^  ?       $96.70. 

11.  A  manufacturer's  sales  for  the  year  amoimted  to  $58,750, 
upon  which  lie  paid  a  government  tax  of  .3^.  What  was  the 
amount  of  the  tax  ? 

13.  A  milliner  pays  a  license  of  $10,  and  her  assessed  income  is 
$835,  on  which  the  tax  is  5^.    How  much  revenue  does  she  pay  ? 

$51.25. 

III.    CUSTOMS. 

434  •  Customs  are  duties  paid  to  Government  on  im- 
ported goods  and  other  property. 

Note. — The  office  at  which  customs  or  duties  arc  collected  is  a  Custom- 
Jlouse  ;  and  a  seaport  town  in  which  a  custom-house  is  situated  is  a  Port  of 
Entry. 

435.  An  Ifivoice  is  a  written  account  containing  a  list 
of  merchandise  sent  to  a  purchaser,  with  prices  and  charges 
annexed. 

In  custom-house  transactions,  certain  deductions  are 
made  on  some  kinds  of  goods,  before  the  duties  are  com- 
puted.   These  are  tare,  leakage,  and  breakage. 

436.  Tat^e  is  a  deduction  made  from  the  weight  of 
goods  sold  in  chests,  boxes,  cases,  casks,  bags,  or  other  envel- 
ope or  covering,  on  account  of  the  vreight  of  such  covering. 

437.  ZfCakage  is  a  deduction  made  from  the  quantity 
of  liquors  imported  in  casks. 


TAXES     AND     DUTIES.  251 

438.  breakage  is  a  deduction  made  from  the  quantity 
of  liquors  imported  in  bottles. 

439.  G7'0SS  }f eight  is  the  entire  weight  of  goods  and 
case  or  covering. 

440.  JVet  Weight  is  the  gross  weight  minus  the  tare. 

441;  JVet  Value  is  the  value  of  goods  at  the  original 
invoice  price,  after  all  deductions  have  been  made. 

442.  Specific  !Duty  is  duty  on  the  number  or  quantity. 

443.  A.d  Ydlorem  ^uty  is  duty  on  the  net  value. 
Note.— A  list  of  rates  of  duties  established  by  Government  is  called  a 


COMI^XJT^TIONS    IN    I3UTIES. 

444.  In  ad  valorem  duties, 

The  net  value  is  the  base  ; 

The  rate  %  of  duty  is  the  rate  ;  and 

The  duty  is  the  percentage.     Hence, 

I.  l^et  value  and  rale  ^f  \  -^  S  ^^^  ^^^  ^^^^  given,  to 
duty  given,  to  find  duly,       )         (find  percentage. 

II.  Specific  duties  are  found  by  midtiplying  the  duly  on  one 
by  the  net  number. 

PM  OBI^EMS. 

13.  The  gross  weight  of  175  boxes  of  raisins  is  33^  lb.  per  box, 
and  the  tare  is  25^.    What  is  the  total  net  weight  ? 

14.  What  are  the  duties,  at  $.25  per  pound,  on  150  chests  of  tea, 
invoiced  at  62  lb.  per  chest  ? 

15.  The  duty  on  opium  is  100^.  What  are  the  custom-house 
charges  on  125  lb.,  invoiced  at  $5.37|-  per  lb.  ? 

16.  A  sugar  refiner  imports  72  hhd.  W.  I.  sugar,  weighing  475  lb. 
each,  and  50  hhd.  molasses  containing  126  gal.  each.  What  are 
the  duties,  sugar  paying  $.03  per  lb.,  and  molasses  $.08  per  gal., 
and  the  tare  on  the  sugar  being  12^^  ?  $l,Ji01.75. 


252  PERCENTAGE. 

JPBOBLJEMS     IN     T  A  X  H:  S     A.  N  D     DUTIES. 

17.  The  valuation  of  the  property  of  a  certain  county  is 
$11,847,500,  upon  which  a  tax  of  $146,909  is  levied.  How  much 
of  this  tax  will  be  paid  by  the  owner  of  a  foundery  which  is  as- 
sessed at  $14,550  ?  $18042. 

18.  A  collector's  fees  for  collecting  a  town  tax  were  $197.73,  and 
the  whole  tax  was-  $14,829.    What  rate  fo  did  the  collector  receive  ? 

mi- 

19.  In  1865,  a  5^  tax  was  required  upon  the  first  $5,000  of  a 
man's  income,  and  10^  upon  all  above  $5,000,  the  exemptions 
being  $600  for  living  expenses,  $200  for  house  rent,  and  the 
amount  paid  for  taxes.  How  much  tax  did  a  man  pay,  whose  in- 
come was  $17,675,  and  who  had  paid  $453  for  taxes?    $1,392.20, 

20.  A  real  estate  agent  who  charged  the  seller  2%.  and  the  buyer 
3;^,  sold  a  house  for  $10,000.    What  was  his  commission  ? 

21.  The  duty  on  tobacco  being  $.35  per  lb. ;  and  on  segars  $3 
per  lb.  specific,  and  50^  ad  valorem ;  what  are  the  duties  on  50 
cases  of  tobacco  invoiced  at  65  lb.  each,  and  175,000  Havana  segars, 
weighing  2,625  lb.,  and  invoiced  at  $45  per  M.  ?  $12,950. 

22.  Upon  the  property  of  a  city  assessed  at  $3,824,600,  a  tax  of 
$72,667.40  is  levied.  Make  a  table,  embracing  the  tax  on  $1  to  $10, 
from  which  the  tax  list  can  be  computed.     See  Manual. 


SECTION  VIII. 
I  jv  T  ^  :r  b  s  t  . 

445.  If  a  person  hires  a  house  or  a  farm,  lie  pays  the 
owner  for  the  use  of  it.  If  a  person  hires  or  borrows  money, 
v/hen  he  pays  the  debt,  he  also  pays  an  additional  sum 
for  the  use  of  the  money.  And  when  a  person  pays  a  debt 
after  it  is  due,  he  pays  an  additional  sum  for  the  credit ; 
i.  e.,  for  the  use  of  the  money  after  the  debt  is  due. 

440.  I7iterest  is  the  sum  paid  for  the  use  of  money. 

447.  ^ri7icipal  is  the  sum  for  the  use  of  which  interest 
is  paid. 


INTEllEST, 


253 


448.  A?nount  is  the  sum  of  principal  and  interest. 

449.  ^ate  per  Cent  per  Aiinum  is  the  interest  on 
$1  for  1  year. 

450.  Simple  Interest  is  the  sum  paid  for  the  use  of 
the  principal. 

451.  Compound  Intei^est  is  the  sum  paid  for  the  use 
of  interest. 

452.  A  Partial  ^ay77ient  is  a  payment  of  a  part  of 
an  obligation  due,  or  that  is  drawing  interest. 

453.  legal  ^ate  is  the  rate  of  interest  allowed  by  law. 

Note. — Any  rate  of  interest  greater  than  tlie  legal  rate  is  Usury. 


454.    TABLE   OF   LEGAL   BATES    OF   mTEREST. 

WHEN    NO    BATE    IS   NAMED. 

KATES    ALLOWED    BY    SPECIAL   CONTRACT. 

H 

La. 

■     Not 
i  exceeding 

TN.     Y.,     Mich.,    Wis., 

Sfo 

Fla.  and  La. 

H 

]Mmn.,S.C.,  Geo.,  Utah, 

r  Ohio,  Iowa,  Miss.,  Ark., 

^  and  Hudson  Co.,  K  J. 

lOfo 

•<  Utah,  and  for  borrowed 

tOfo 

Ala.  and  Tex. 

.(  money  in  Mich,  and  111. 

j  Cal.,    Or.,   Kan.,   Neb., 
1  W.  T.,  Nev.,  and  Col. 

12fo 

j  Wis.,  Tex.,  and  on  judg- 
(  ments  in  Minn. 

r  All    the    other   States, 

15fo 

Neb. 

Qfo 

)  D.  C,  and  bank  inter- 

20fo 

Kan. 

(  est  in  La.  and  Kan. 

Any  rate 

per  cent 

agreed 

upon. 

(  R.  I,  Minn.,  Cal.,  W.  T., 
1  Nev.,  and  Col. 

c-a.se  I. 
Computations  of  Simple  Interest. 

455.  In  all  the  previous  Sections  of  this  Chapter,  rate  % 
is  a  fixed  sum  without  regard  to  time.  But  in  interest,  the 
entire  rate  per  cent  paid  on  $1,  depends  upon  the  time. 
Thus,  if  the  rate  is  6%  per  annum,  the  rate  %  on  $1  for  1 
year  is  .06  ;  for  3  years  it  is  3  times  .06,  or  .18 ;  for  6 
months  or  J  year  is  ^  of  .06,  or  .03,  etc.     Hence, 

The  rate,  or  the  percentage  on  $1,  is  the  product  of  the  rate 
per  cent  per  annum  and  the  time  expressed  in  years. 


254  PERCENTAGE. 

I.    GENERAL   METHOD. 

456.  Ex.  What  is  the  interest  of  $287.50  for  3  years,  at 

Explanation. — Since  the  rate  is  1%  solution. 

per  annum,  the  interest  for  1  year        $28  7.5  0  Principal 

is  .07  times  the  principal,  and  the        ^_^  ''^«^^- 

interest  for  3  years  is  3  times  the        $20,125  int. /or  i  yr. 

interest  for  1  year.     We  therefore ^ 

multiply  $287.50  by  .07,   and  the       $  6  0, SI  5  int.  for  ^yr. 

product,  $20,125,  by  3.     The  final 

result,  $60. 37^,  is  the  required  interest.     Hence, 

Interest  for  years  is  the  product  of  principal,  rate,  and  time. 

FMOBIjEIIS.       * 

1.  What  is  the  interest  of  $515.50  for  1  year,  at  %  ?      $80.93. 

2.  What  is  the  amount  of  $325  for  1  year,  at  6^  ?     (See  44§.) 
8.  What  is  the  interest  of  $117.35  for  2  years,  at  5^  ?    $11.72^. 

4.  If  I  borrow  $390  for  4  years,  at  7^,  what  amount  will  be  due 
at  the  expiration  of  the  time  ?  $499.20. 

5.  What  is  the  interest,  and  what  the  amount,  of  $1,068.50  for  1 
year,  at  8^  ?  Amount,  $1, 153.98. 

457.  Ex.  What  is  the  in-  soltttion. 
terest  of  $654.75  for  1  jr.  5              $  6  5  4.7  5  Principal. 

.0  6  Pate. 


Explanation. — Since   1   yr.  $3  9.2  8  5  0  int. for  i  yr. 

5  mo.,  or  17  mo.,  is  \%  yr.,  we ^ 

first  find  the  interest  for  1  yr.,  2  7^995 

as  in  (456) ;  and  then  multiply  39285 

this  interest,  $39,285,  by  |i,  $6  67,8  45^12 

the  required  time.     That  is,  $55,65+  S^'.forwyr., 

when  there  are  months  in  the  <  oriyr.^  mo. 

given  time,  we 

Multiply  the  interest  for  1  year  hy  the  number  of  months, 
and  divide  the  product  by  12. 


INTEREST.  255 

Notes. — ^1.  In  computations,  the  partial  results  should  be  carried  to  four 
decimal  places. 
2.  In  final  results,  if  the  mills  are  5  or  more,  it  is  customary  to  call  them 

1  cent,  and  if  they  are  less  than  5  to  reject  them.    (Sce^  163.) 

BJtOBLEMS. 

6.  What  is  the  interest  of  $2,160  for  1  year  3  months,  at  7^  ? 

7.  What  is  the  interest  of  $39.25  for  3  yr.  8  mo.,  at  5^? 

8.  What  is  the  amount  of  $1,278  for  11  mo.,  at  7^  ?  $1,360.01. 

9.  rind  the  interest  of  $9,500  for  3  yr.  1  mo.,  at  4^. 

10.  How  much  interest,  at  8^,  must  I  pay,  for  the  use  of  $2,575 
from  May  11,  1868,  to  Sept.  11,  1869  ?  $27Jt.G7. 

458.  Ex.  What  is  the  solution. 
interest   of  $761.25  for  2        ^^  yr.  5  mo.  16  da.  =  29.51,-  mo, 

yr.  5  mo.  16  da.,  at  8^  ?  $161.25  Principal. 

Explanation.  —  Since  30  .OSjiate. 

days  are  1  month,  every  3  $ 6  0.9  0  00 

days  are  1  tenth  of  a  month ;  2  9.5  j- 

16  days  are  J/  tenths  or  20  3 

5i  tenths  of  a  month  ;  and  nVgY 

2  yr.  5  mo.  16  da.  are  29.5|  1218 

29  5'  ~ — 

mo.  or  -^^  yr.   We  there-       $1798.58    { 1 2 

12  <^  1  !  Q  Si  fi^\  Int.  for  2  yr. 

fore  multiply  the  interest  ^-/-^^.oo^^  ^mo.i&da. 

for  1  year  by  the  number  of  months,  and  divide  the  product 

by  12,  as  in  457. 

PJIOBI^EMS. 

11.  What  is  the  interest  of  $198.50  for  4  mo.  9  da.,  at  4^? 

12.  What  is  the  interest  of  $10,796  for  2  yr.  1  mo.  24  da.,  at  7^  ? 

13.  Fmd  the  amount  of  $18,450  for  1  mo.  15  da.,  at  ^'/o. 

14.  How  much  interest,  at  6^,  has  accrued  on  a  note  for  $94.75, 
that  has  been  due  3  yr.  2  mo.  6  da.  ?  $18.10. 

15.  What  is  the  amount  of  $978.18  from  Sept.  24,  1867,  to  Oct. 
25,  1869,  at  8^?  $1,U1.J^. 


25G  PERCENTAGE. 

II.    SIX   PER    CENT    METHOD. 

459.  The  interest  of  $2  for  1  year,  or  12  montlis,  at  Q%,  or 
of  $1  for  the  same  time,  at  12%,  is  $.12.     Hence, 

I.  The  interest  of  any  sum  at  6%  is  the  same  as  the  interest  of 
one  half  that  sum  at  12%. 

II.  At  12%o  per  annum,  the  rate  is  1%)  per  month. 
Ex.  Find  the  interest,  at  Q%,  solution. 

of  $52.69  for  2  yr.  3  mo.  18  da.     2  yr.  Smo.18  da.  =27.6  mo. 

Explanation.— We  first  di-  S  S  2.6  9  \2 

vide  the  principal,  $52.69, by  ^26.3^5  1  the phu. 

2,  to  find  the    sum   on  which  ^2  7  6  Rate  at  \%  per  mo. 

to  compute  interest  at  12^  i  5  8  0  7  0 

(I.).     We  then  multiply  this  18J^Jj.l5 

result,  $26,345,  by  the  rate  52690 

at  1%)  per  month,  which  is       $7.2  71220  interest. 
.01  of  the  time  expressed  in 

months  (H.).     The  final  result,  $7.27,  is  the  required  in- 
terest. 

ph  OB  lems. 

16.  How  much  interest,  at  6^,  will  be  due  in  5  yr.  on  a  loan  of 
$5,790?  $1,737. 

17.  What  is  the  interest  of  $728.18  for  1  yr.  11  mo.,  at  6^? 

18.  If  $2,765  be  placed  at  interest  at  6^,  Mar.  14,  1869,  what 
will  be  due  Dec.  13,  1870  ?  $3,054.86. 

19.  What  is  the  interest  of  $20  for  12  yr.,  at  5^  ? 

(6^  — i  of  itself  =5^.)  $12. 

20.  Find  the  amount  of  $417.61  for  3  yr.  7  mo.,  at  8fc. 

(6^  +  1  of  itself  =8^.)  '    $537.32. 

III.   SEVEN  PER  CENT  METHOD,  FOR  DAYS. 

460.  Computing  interest  on  the  basis  of  30  days  to  a 
month,  gives  360  days  to  a  year.     Hence, 

I.  The  product  of  any  principal  multiplied  by  any  given 
number  of  days  expressed  as  hundredths,  is  the  interest  at  360% 
per  annum,  or  1%)  per  day. 


375 
750 

$7,8  7  5  \e 

$1,3125  \  6 
,21875 

INTEREST,  257 

n.  The  product  of  any  principal  multiplied  by  any  given 
number  of  days  expressed  as  thousandths,  is  the  interest  at 
per  annum,  or  .l%per  day. 

III.  The  interest  at  36%  divided  by  6  gives  the  interest  at  t 

IV.  Interest  at  Gfo  plus  interest  at  1%  equals  interest  at  7%, 

Ex.  What  is  the  interest  of  $125  for  63  days,  at  1%  ? 

ExpL.iNATioN. — We    first    multiply  solution. 

the  principal,  $125,  by  .063,  and  ob-  $12  5  Pnn. 

tain  $7,875,  the  interest  at  36^  (n.).  -^^^ 

We  divide  this  result  by  6,  and  ob- 
tain $1.31|,  the  interest  at  6%  (HI.). 
We  then  divide  this  result  by  6,  and 
obtain  the  interest  at  1% ;  and  add- 
ing the  last  two  results,  we  have 
.$1.53 1,  the  required  interest  (IV.).  $  1.53  12  5  int. 

PROBLEMS. 

21.  What  is  the  interest  of  $735  for  27  days,  at  Ifo  ?  •     $3.86, 
23.  What  is  the  amount  of  $250  from  Jan.  18  to  March  30,  1868, 
at  7^  ?  $253.50. 

23.  Dec.  24,  1868,  I  borrowed  $25.50,  and  paid  it  June  1,  1869, 
with  7^  interest.     What  amount  was  due  ? 

24.  What  is  the  interest  of  $45.75  for  90  da.,  at  7^  ?  $.80. 

25.  What  is  the  amount  of  $1,250  for  63  da.,  at  7j^  ?     $1,265.31, 

461.  ^ules  for  Computi7ig  Interest, 

I.  General  Method. 

1.  For  1  year.  Multiply  the  principal  by  the  rate. 

2.  For  2  or  more  years.  Multiply  the  interest  for  1  year  by 
the  number  of  years. 

3.  For  any  other  time.  Multiply  the  interest  for  1  year  by 
the  time  expressed  in  months  and  tenths  of  a  month,  and  divide 
the  product  by  12. 


258  TERCENTAGE. 

II.  Six  Per  Cent  Method. 
Divide  the  principal  by  2,  and  multiply  the  quotient  by  .01 
of  the  time  expressed  in  months. 

ni.  Seven  Per  Cent  Method,  for  Days. 

1.  Midtiply  the  principal  by  .001  of  the  number  of  days. 

2.  Divide  the  product  by  6,  and  to  the  quotient  add  -^  of 
itself. 

Notes.— 1.  360  days  +  /^  of  360  days  (5  days)  =  365  days.  Hence,  if 
interest  for  days  is  required  at  365  days  to  a  year,  subtract  from  itself  ^\^  of 
the  interest  found  by  tlie  "1%  method. 

2.  To  find  the  amount,  we  may  first  find  the  amount  of  $1  at  the  given 
rate  for  the  given  time,  by  any  one  of  the  above  rules,  and  then  multiply 
the  principal  by  this  amount. 

3.  The  months  and  days  may  be  reduced  to  the  decimal  of  a  year  (sec 
356,  II.),  and  the  interest  for  1  year,  at  the  given  rate  ^,  may  then  be  multi- 
plied by  the  time  expressed  in  years  and  decimals  of  a  year. 

r  II  o  B  Ij  ems. 

26.  What  interest  must  I  pay  for  the  use  of  $378.64  for  3  years, 
at  7^? 

27.  What* is  the  amount  of  $473  for  7  yr.  7  mo.,  at  5^ ? 

28.  What  is  the  interest  of  $419.84  for  1  yr.  11  mo.  18  da.,  at 
5^?  $Jtl.28. 

29.  A  debt  of  $1,560  was  contracted  May  23,  1868.  How  much 
was  due  June  18,  1869,  interest  at  6^  ?  $1,660.10. 

30.  The  balance  due  on  a  mortgage,  Nov.  20,  1868,  was  $3,750. 
What  was  the  amount  due  Aug.  20,  1869,  interest  7^  ? 

31.  What  is  the  amount  of  $75  for  8  yr.,  at  10^  ?  $135. 

32.  A  note  for  $1,116,  bearing  date  Albany,  N.Y.,  Oct.  9,  1866, 
was  paid  Oct.  9,  1869,  with  interest.     What  amount  was  paid  ? 

33.  What  amount  was  due  Aug.  5,  1869,  on  a  note  for  $1,650, 
dated  Philadelphia,  Dec.  5,  1867  ?  $1,815. 

34.  I  bought  a  house  and  lot  in  Cleveland,  for  $4,750,  paying 
$2,000  down,  and  giving  a  mortgage  for  the  balance,  due  in  3  years. 
What  was  the  amount  of  the  mortgage  when  due  ?  $3,!245. 

35.  How  much  was  due.  May  3, 1869,  on  a  note  for  $2,860.  dated 
San  Francisco,  July  3,  1867  ? 


INTEREST.  259 

36.  What  is  the  amount  of  $743.18  for  1  yr.  10  mo.  12  da.,  at  8^  ? 

37.  If  I  borrow  $12,500  in  New  Haven,  Conn.,  and  loan  it  in 
New  York,  how  much  do  I  gain  in  1  yr.  7  mo.  ?  $197.92. 

38.  If  I  loan  $1,500,  at  7^,  Aug.  17,  1869,  how  much  will  be 
due  June  30,  1871  ?  $1,696.29. 

39.  What  is  the  interest  of  $10  for  15  yr.  4  mo.,  at  6^  ? 

40.  A  note  for  $293,  dated  Detroit,  Apr.  26,  1867,  was  paid  Jan. 
26,  1869.     What  was  the  amount  paid  ?  $328.89. 

41.  At  7^,  what  is  the  amount  of  $73.49  from  Nov.  27,  1867,  to 
Feb.  7,1870?  $8J^.78. 

42.  A  man  bought  a  farm  in  Minnesota  for  $2,280,  paying  $1,000 
down,  and  the  balance  in  10  months,  with  interest.  How  much  was 
the  last  payment  ?  $1,354.67. 

43.  Find  the  amount  of  $856.75  for  2  years,  at  5fo.     $942.42-1. 

44.  How  much  interest  will  I  have  to  pay  on  a  loan  of  $7,650 
for  20  days,  at  7^  ?  $29.75. 

45.  What  is  the  amount  of  $25,390  for  7  months,  at  10^  ? 

40.  Jan.  10,  1869,  I  borrowed  $1,280  in  Hartford,  Conn.,  and 
paid  it  Aug.  7,  1869,  with  interest.      How  much  did  I  pay  ? 

47.  What  is  the  interest  of  $1,310  for  1  yr.  1  mo.,  in  Ya.  ? 

48.  How  much  will  be  due  June  19,  1871,  on  a  note  for  $1,750, 
dated  Boston,  June  19,  1869,  with  interest  ?  $1,960. 

49.  Find  the  amount  due  Mar.  17,  1870,  on  a  note  for  $217.85, 
dated  St.  Louis,  Sept.  17,  1867,  with  interest. 

50.  A  man  who  is  paying  $375  a  year  for  house  rent,  borrows 
$5,000,  at  6^,  with  which  he  buys  the  house.  Does  he  gain  or 
lose  by  the  transaction  ?  JSb  gains  $75  per  annum. 

CA.SE     II. 
Compound  Interest. 

462.  In  computing  compound  interest, 

I.  Hie  amount  of  the  principal  for  1  year  is  (he  principal 
for  the  second  year,  the  amount  of  this  principal  for  1  year  is 
the  principal  for  the  third  year,  and  so  on. 

n.  The  final  amount  minus  the  principal  is  the  interest. 


2G0  PERCENTAGE. 

Ex.  What  is  the  amount  of  $127.50  at  compound  interest 
for  2  jr.,  at  6%  ?     What  is  the  interest  ? 

Explanation.  —  Since   the  solution. 

amount  is  the  product  of  the  $12  7.50  Pnn. 

principal    multiphed    by    1  1-0  Q     i  +  »*«^«- 

plus  the  rate  (see  461,  Note  11 6  50 

2),  we  multiply  the  principal,  127  5 

$127.50,  by  1.06,  and  obtain  $18  5.1 5\  p^I.-^Z/o^^vC 

$135.15,   the   amount  for  1  1-0  ^   i+rate. 

year.       We     multiply    this  8109  0 

amount  by  1.06,  as  before,  13  515 

and  obtain  $143,259,  the  re-  $  IJf  8.2  5  9    Amt.for  2  yr. 

quired  amount  for  2  years.  12  7.5  0       Prin. 

Then,  subtracting  the  prin-  $15,759    int. 
cipal,     $127.50,     from    this 
amount,  we  have  $15.76,  the  required  interest. 

Note. — When  interest  is  due  semi-annually,  quarterly,  or  monthly,  the 
amount  of  the  principal  for  the  fixed  period  of  time  is  the  principal  for  the 
next  period. 

FJIOBJLEMS. 

51.  What  is  the  amount  of  $731.45  for  3  years,  at  6^  compound 
interest  ?  $859.26. 

53.  What  is  the  compound  interest  of  $75.50  for  2^  years,  at  6^, 
payable  semi-annually  ?  $12.03. 

53.  The  principal  is  $35.75,  the  time  4  years,  and  the  rate  7^ 
compound  interest.     WTiat  is  the  amount  ? 

54.  How  much  will  $535  amount  to  in  1^  years,  at  7^  compound 
interest,  payable  semi-annually  ?  $582.08. 

55.  What  is  the  compound  interest  of  $437.50  for  1  yr.  3  mo.,  at 
Qfc,  payable  quarterly  ? 

56.  At  34|^,  interest  payable  monthly,  what  is  the  compound 
interest  of  $575  for  3-i-  yr.  ? 

57.  What  is  the  difference  between  the  simple  and  the  compound 
interest  of  $5,435  for  4  years,  at  Qfo  ?  $121.94. 


INTEREST. 


261 


C^SE    III. 
Partial  Payments. 

463.  When  partial  payments  are  made  upon  notes,  bonds, 
mortgages,  or  other  obHgations  bearing  interest,  the  U.  S. 
Courts  have  estabhshed  the  following  principles  : 

I.  Paijments  must  he  applied  in  the  first  place  to  the  dis- 
charge of  interest  due,  and  the  balance  toward  the  discharge  of 
the  principal. 

II.  Interest  must  not  he  added  to  the  principal  so  as  to  draw 
interest. 

m.  The  principal  must  remain  unaltered,  when  a  payment 
is  less  than  interest  due. 

Ex.  A  note  for 
S960,  at  Q%  interest, 
was  given  Apr.  10, 
1868.  A  payment  of 
$225  was  made  Jan. 
19, 1869,  and  another 
of  $25,  Nov.  3,  1869. 
What  amount  was 
due  Jan.  5,  1870  ? 

Explanation.  — We 
first  find  the  amount 
of  the  principal  from 
the  date  of  the  note 
to  Jan.  19,  1869,  the 
time  of  the  first  pay- 
ment (see  461, 11.),  to 
be  II,  0  04.  64.  We 
next  subtract  the  pay- 
ment, $225,  from  this 
amount  (I.),  and  have 
a  remainder  of  $779.- 
64  for  a  new  princi- 
pal.   Since  the  pay- 


BOLTJTIOX. 

$9  60  Prin.  12 

SJ^80 
,0  9  3 

432 

$    JfJf.6  Jf.  Int.  to  Jan.  19,  1869. 
9  60           Prin. 

$  10  0  4.6  Jf  Amt.  due  Jan.  19, 1869. 
22  5          Payment,     "           " 

$T  19.6  4  New  Prin.   \2 

$  3  8  9.82 
.115i 

12994 
194910 
38982 
38982 

S     44.95  92  4  Int.  to  Jan.  5, 1870. 
7  7  9.64               Prin. 

$824.59924  Amt.  to  Jan.  5,  ISTO. 
2  5                      Payment,  Nov.  8, 1869. 

$  7  99.6  0              Ami.  due  Jan.  5, 1870. 

262  PERCENTAGE. 

ment,  $25,  made  Nov.  3,  1869,  did  not  exceed  the  interest 
due  (in.),  we  find  the  amount  of  $779.64  from  Jan.  19, 
1889,  to  Jan.  5,  1870.  Then,  subtracting  from  this  amount, 
$824.59924,  the  payment  of  $25,  made  Nov.  3,  1869,  we  have 
$799.60,  the  required  amount  due  Jan.  5,  1870.     Hence, 

461.  (Rule  for  Computinff  Interest  191  I*a7^tlal  'Paymejits, 

I,  From  the  amount  of  the  principal  computed  to  the  time 
when  the  payment  or  sum  of  the  payments  equals  or  exceeds  the 
interest  due,  subtract  the  payment,  or  sum  of  the  payments. 

n.  The  remainder  is  a  new  principal,  with  which  proceed  as 
before. 

JPM  O  B  LE 31 S . 

58.  On  Nov.  5,  1867,  the  face  of  a  mortgage  on  a  farm  in  Mich, 
was  $2,875,  and  $1,000  was  paid  Aug.  23,  1868.  What  amount 
was  due  May  17,  1869  ?  $2,140.51. 

59.  A  mechanic  bought  a  house  and  lot  in  Salem,  Mass.,  for 
$1,750,  paying  $500  down.  One  year  afterward  he  paid  $387.50. 
How  much  then  remained  due  ?  $937.50. 

00.  Upon  a  note  for  $765,  dated  Buffalo,  N.  Y.,  Mar.  14,  1867, 
there  was  paid,  Oct.  31,  1868,  $50 ;  and  June  11,  1869,  $285.  The 
note  was  taken  up,  Sept.  25,  1869.     How  much  was  then  due  ? 

61.  May  7,  1867,  a  capitalist  loaned  $10,000,  at  ^fc.  Dec.  28, 
1867,  $4,800  was  paid ;  and  July  14,  1868,  $3,750.  What  sum  was 
due  Jan.  3,  1869  ?  $2,020.0^. 

VT.,    N.   H.,    AND    CONN.    RULES. 

465.  In  Vt.  and  N.  H.,  a  written  stipulation  to  pay  inter- 
est annually,  allows  the  creditor  simple  interest  on  interests 
due  on  the  principal  and  remaining  unpaid  after  the  end  of 
each  year.  This  allowance  of  simple  interest  for  the  use  of 
interest  due,  is  called  Annual  IntereM.     see  Manual. 

The  differences  between  the  U.  S.  Court  Eule  for  com- 
puting interest  in  Partial  Payments,  and  the  rules  in  Vt., 
N.  H.,  and  Conn.,  are  as  follows  : 


INTEREST.  263 

466.   The  Vermont  ^ule, 

I.  Simple  interest  is  allowed  on  all  unpaid  annual  inter ests, 
from  the  time  they  become  due  to  the  time  of  final  settlement. 

II.  Simple  interest  is  allowed  on  all  payments,  from  the  time 
they  are  made  to  the  end  of  the  year,  or  to  the  time  of  final 
settlement, 

467.    T/ie  JVew  Ilai7ipshire  'Eute, 

I.  Interest  is  allowed  on  payments  only  when  a  payment,  or 
the  sum -of  two  or  more  payments,  exceeds  the  interest  due. 
n.  In  all  other  rejects  the  rule  is  the  same  as  in  Vermont. 

Notes. — 1.  In  computing  annual  interest  in  these  States,  the  computa- 
tions must  be  made  for  intervals  of  1  year,  or  to  the  time  of  final  settle- 
ment, when  that  occurs  within  a  year. 

2.  The  interest  in  partial  payments  is  computed  by  the  U.  S.  Court  Rule, 
unless  annual  interest  is  stipulated  in  the  note  or  agreement. 

468.   27ie  Connecticut  ^ute, 

I.  Interest  is  allowed  on  payments  to  the  end  of  a  year,  when 
they  exceed  the  interest  due,  at  the  time  they  are  made  ;  or  to 
the  time  of  settlement,  when  that  occurs  within  a  year. 

n.  When  more  than  a  year  passes  from  the  date  of  any  com- 
putation, without  payments  being  made,  interest  is  computed  for 
the  whole  time,  by  the  U.  S.  Court  Rule. 

PJt  OB  TjEMS. 

62-65.  Find  the  amount  due  Oct.  1,  1869,  on  the  following  note, 
computing  the  interest  by  the  U.S.,  the  Vt.,  the  N.  H.,  and  the 
Conn.  Rule. 

^/,<rSd  QJ^a^^e  /,    /(T/f/. 

C/n/  </e?nanc/^    o^  /lionncie    ^a  X'^  *^    veaiei  Gta^^ee?v     awunmeix 

Indorsements  on  the  back  of  this  note :  Oct.  17,  1867,  $350 ; 
Feb.  33,  1868,  $100 ;  Dec.  30,  1868,  $50 ;  July  17,  1869,  $335. 

U.  S.  Mule,  $1,U9.U;  yt',  $hW'S2; 
N.  H.,  $XU6.77;  Conn,,  $1,U6.0S. 


264 


PERCENTAGE. 


Converse  Operations  in  Interest. 

460.  In  computations  in  interest, 

Principal  is  the  base  ; 

The  product  of  rate  %  per  annum  and  time  expressed  in 
years,  is  the  rate  ; 

The  interest  is  the  percentage  ;  and 

Interest  is  the  product  of  the  three  factors,  principal, 
rate,  and  time.     Hence, 


I.  Principal^    rate    and) 
time  given,  to  find  interest,  ) 

II.  Interest,  rale  and  time  ]^ 
given,  to  find  principal,         ) 

m.  Principal,  rate  and ) 
time  given,  to  find  amount,  ) 

IV.  Amount,     rate    and ) 
time  given,  to  find  principal, ) 

V.  Principal,  interest  and 
time  given,  to  find  rate,  and 

YI.  Principal,      interest 
and  roie  given,  to  find  time,^ 


are-^ 


j      Base  and  rate  given,  to 

I  find  percentage. 

j      Percentage  and  rate  giv- 

i  en,  to  find  base. 

j      Base  and  rate  given,  to 

I  find  amount. 

j      Amount  and  rate  given^ 

(to find  base. 

Base,  percentage,  and  one 
of  the  two  factors  of  rate 
given,  to  find  the  other  fac-- 
tor. 


Note. — In  V.  and  VI.,  the  product  of  three  factors  (interest),  and  two  of 
the  factors  (principal  and  rate,  or  principal  and  time)  are  giyen,  to  find  the 
third  factor. 

J*JJ  OBIjEMS. 

66.  What  is  the  interest  of  $580  for  11  mo.  27  da.,  at  6^  ? 

67.  What  is  the  interest  of  $119.50  for  1  yr.  7  mo.  18  da.,  at 

68.  The  interest  is  $56,  the  rate  .06,  and  the  time  2  mo.  10  da. 
What  is  the  principal  ?  $1^,800. 

09.  I  loaned  a  certain  sum  of  money,  at  7^,  and  received  $98.28 
as  the  interest  for  1  yr.  8  mo.  24  da.   What  was  the  sum  loaned  ? 

70.  What  is  the  amount  of  $7,365  for  2  yr.  4  mc,  at  5^  ? 

71.  What  is  the  amount  of  $390  for  7  mo.  10  da.,  at  H  % 


DISCOUNT.  265 

73.  What    principal    put    at    interest  at  6^,  will   amount  to 
$1,073.10  in  1  yr.  7  mo.  ?  $980. 

73.  What  principal  put  at  interest  at  4^,  will  amount  to  $183.20 
in  5  yr.  4  mo.  13  da.  ? 

74.  The  interest  is  $118.30,  the  principal  $3,800,  and  the  time 
5  mo.  19  da.    What  is  the  rate  ? 

75.  At  what  rate  will  $508.50  earn   $89,609  in   3   yr.   3  mo. 
13  da.  ?  '  8^' 

76.  In  what  time  will  $475  earn  $71.25,  at  6^  ? 

77.  The  principal  is  $684,  the  interest  $103.68,  and  the  rate  10^^. 
What  is  the  time  ? 

78.  What  is  the  interest  of  $3,750  for  1  yr.  1  mo.  13  da.,  at  l^  ? 

79.  What  prmcipal,  at  5^,  will  produce  $19.09  interest  in  3  yr. 
3  mo.  18  da.  ?  $166. 

80.  The  interest  of  $6,000  is  $805,  for  1  yr.  11  mo.     "What  is  the 
rate  ?  7^. 

81.  I  paid  $165.37^  for  the  use  of  $1,350,  at  I'/o.     What  was  the 
time  ?  1  yr.  9  mo. 

83.  The  principal  is  $388,  the  rate  7^,  and  the  time  3  yr.  4  mo. 
10  da.    What  is  the  amount?  $355.76. 

83.  The  principal  is  $19,600,  the  amount  $31,047.95,  and  the  rate 
4|^.     What  is  the  time  ?  ,  1  yr.  7  mo.  21  da. 

84.  New  York,  Jan.  35,  1869, 1  paid  $169.85,  the  amount  due  on 
a  note  given  Sept.  15,  1865.     What  was  the  face  of  the  note  ? 

85.  At  what  rate  will  $560  amount  to  $659.40,  in  3  yr.  11  mo. 
15  da.  ?  G'fo. 


SECTION    IX. 
DISCOZTjYT, 

470.  discount  is  a  sum  deducted  for  the  payment, 
before  it  becomes  due,  of  a  note  or  other  debt  not  drawing 
interest. 

471.  The  JF'ace  of  an  obligation  is  the  sum  to  be  paid 
when  the  obligation  is  due,  or  At  3Iaiurity. 

13 


266  PERCENTAGE. 

472.  "Pt^esent  JForth^  or  Proceeds ,  is  the  face  of  an 
obligation  minus  the  discount. 

473.  Connnercial  'Discount,  or  Per  Ce7it  Off,  is  a 

deduction,  from  the  face  of  the  obligation,  of  some  %  agTeed 

upon,  without  regard  to  time. 

Note. — "When  the  sum  deducted  depends  upon  both  rate  and  time,  it  is 
sometimes  called  True  Discount. 

COnVIFXJTJ^TIONS    IN^    IDISCOTJT^T. 

474.  In  Computations  in  Discount, 
Present  worth  or  proceeds  is  the  base ; 

The  product  of  the  rate  per  cent,  per  annum  and  the  time 
expressed  in  years,  is  the  rate  ; 

The  face  of  the  obligation  is  the  amount ;  and 
The  discount  is  the  percentage.     And 

475.  In  Commercial  Discount,  or  Per  Cent  Off, 
Invoice  price,  or  the  face  of  an  obligation,  is  the  base  ; 
Kate  %  off  is  the  rate  ;  and 

The  commercial  discount  is  the  percentage.     Hence, 

I.  Invoice  price  or  face,  \         ( 

and  rate  %  off  given,  to  find}  ^  \      ^"'^  "'"^  '''^  ^''""'  '" 
commercial  discount,  )         (lo  find  percentage. 

II.  Face,  rate  %  per  an-  )         ( 

num  and  time  given,  to  find\  «  ]     ^'"o^"'  «"<^  ^'^  given, 
proceeds,  )         (to  find  base. 

PHOBIC  JS  MS. 

1.  What  is  the  commercial  discount  on  a  bill  of  goods  invoiced 
at  $375.75,  sold  on  3  mouths'  time,  at  2^fo  off  for  cash  ?       $9.39. 

2.  If  I  buy  a  bill  of  goods  amounting  to  $23.7.50,  on  30  days,  3^ 
off  for  cash,  what  is  the  commercial  discount  ? 

3.  A  merchant  buys  a  bill  of  goods  amounting  to  $1,302.40, 
on  3  months'  time,  and  is  allowed  5fo  off  for  cash.  What  sum  does 
he  pay?  $1,237.28. 

4.  When  money  is  worth  6^  per  annum,  how  much  must  be  dis- 
counted for  the  present  payment  of  a  note  for  $375.70,  due  in 
8  months?  %U.J^. 


GOVERNMENT    SECURITIES.  267 

5.  What  is  the  present  worth  of  a  note  for  $304.50,  due  in  1  yr. 
3  mo.,  when  money  is  worth  l^o  ?  $280. 

6.  Find  the  present  worth  of,  and  the  discount  on,  $56,  due  in 
7  mo.  6  da.,  in  New  York. 

7.  A  merchant  buys  a  bill  of  goods,  invoiced  at  $975,  on  60  days. 
If  5^  off  is  allowed  for  cash,  how  much  will  he  gain  by  borrowing 
the  money  at  7^^,  and  cashing  the  bill  ?  $37,944. 

8.  A  farmer  bought  a  horse  for  $140,  giving  his  note  due  in 
1  year.  4  mo.  24  da.  afterward  he  paid  the  note,  the  holder  allow- 
ing discount  at  7fc.     How  much  did  he  pay  ? 

9.  Which  is  the  greater,  the  interest  or  the  discount  of  $1,712  for 
1  year,  at  7^,  and  how  much  ?  The  interest^  $7.84. 

10.  Sold  a  bill  of  goods  amounting  to  $1,260,  on  4  months,  and 
made  the  buyer  the  customary  discount  of  5^  off  for  30  days,  and  a 
further  discount  of  lO^'^  off  for  cash.  What  were  the  cash  proceeds 
of  the  sale?  $1,077.30. 

11.  An  invoice  of  books,  at  retail  prices,  amounts  to  $920,  the 
commercial  discount  is  25^  off,  and  2^fo  off  for  cash.  What  are  the 
net  proceeds  of  the  invoice  ?  $672.75. 

12.  What  is  the  difference  between  discounting  a  bill  of  goods 
at  30^  and  5^  off,  and  discounting  the  same  bill  at  5^  and  20^  off? 


SECTION  X. 

GO  r£^^JVM^J\r  T   S  BCT/^I  TIBS  , 

476.  A  ^ond  is  a  written  obligation  from  one  person  or 
party,  securing  to  another  the  payment  of  a  given  sum  at 
a  specified  time. 

477.  Whenever  the  U.  S.  Government  borrows  money,  it 
gives  to  the  lender  a  bond  for  the  sum  borrowed,  with 
interest  payable  semi-annually,  or  annually. 

478.  Governme7it  Securiiies  are  bonds,  or  certifi- 
cates of  indebtedness,  of  the  Government,  to  the  holder  of 
the  same. 


268  PERCENTAGE. 

479.  The  principal  U.  S.  Securities  are  7-30's,  5-20's, 
10-40's,  5's  of  71,  5's  of  74,  6's  of  '81,  and  the  U.  S.  Pacific 
R.  R  Currency  6's  of  '95,  '96,  and  subsequent  dates. 

The  rates  of  interest  on  these  securities  are,  on 
The  T-SO^S^  ^fm%^  payable  in  legal-tender  notes  ; 
The  5-20' s,  and  T?ie  6's  of '87,  6%,  payable  in  gold ; 
The  W'AO'S,  5%,  payable  in  gold  ; 
The  IZ.  S,  "Pacijic  ^,  7i,  Ctirre7icy  6's,  Qfc,  payable 
in  currency. 

Notes.— 1.  The  interest  on  Goyernment  bonds  for  small  denominations, 
as  $50  and  $100,  is  payable  annually,  as  was  also  the  interest  on  the  7-30's. 

2.  The  5-20'3  can  be  paid  at  any  time  from  5  to  20  years  from  date,  at  the 
option  of  the  Government.    They  were  issued  in  1862,  '64,  '65,  '67,  and  '68. 

3.  The  5-20's  of  '67,  '68,  were  issued  in  those  years,  to  take  up  the  7-30's. 

4.  The  10-40's  can  be  paid  at  any  time  from  ten  to  forty  years  from  date, 
at  the  option  of  the  Government.    They  v/ere  issued  in  1864. 

5.  The  U.  S.  Pacific  R.R.  Currency  6's  were  issued  in  1865  to  1869  inclu- 
sive, and  are  due  in  from  1895  to  1899  inclusive,  according  to  date  of  issue. 

SEICXJRITIIGS. 

480.  The  par  value  or  face  of  the  bond  is  the  base  ; 
The  rate  %  of  premium  or  discount  is  the  rate  ;  and 
The  market  value  is  the  amount  or  difference.     Hence, 

I.  Face  of  hoAd  and  rate  )  •  (  Base  and  rate  given,  to 
given,  to  find  market  value,    )         \  find  amount  or  difference, 

II.  Market  value  and  rate  )  4„  j  Amount  or  difference  and 
given,  to  find  face  of  hond,  )         \  rate  given,  to  find  base. 

m  OBLJEMS. 

1.  How  much  will  3  lOOO-dolIar  7-80  bonds  cost,  at  3|-^  premium  ? 

2.  If  I  sell  $21,500  of  Missouri  C's,  at  32^  discount,  how  much 
shall  I  realize  ?  $lJf,,  620. 

3.  When  gold  is  worth  124,  what  amount  of  currency  can  be 
bought  for  $5,400  m  gold  ? 

4.  A  capitalist  invested  $10,176  in  New  York  City  bonds,  at  4^ 
discount     What  amount  in  bonds  did  he  receive?  $10,600. 


BANKING. 


269 


5.  What  is  the  gold  value  of  $5,485  of  currency,  when  gold  is 
137|? 

6.  "When  gold  is  140,  which  is  the  better  investment,  7-30's  at 
102,  or  Qfc  5-20's  at  108  ? 

7.  A  broker  bought  5  800-dollar  6fo  county  bonds,  at  3*^  dis- 
count, and  afterward  sold  them  at  5^  discount.  How  much  did  he 
lose  by  the  transaction  ?  $22.50. 

8.  If  I  invest  $5,400  in  Pacific  R.R.  6's,  at  28^  below  par,  what 
will  be  the  annual  interest  due  me?  The  interest  will  be  what 
rate  per  cent  upon  the  investment?  Int.  $4^0;  rate,  8.0. 

9.  A  capitalist  invests  $20,500  in  U.  S.  10-40's,  at  2^fo  premium. 
If  gold  is  worth  135,  what  fo  in  currency  does  he  receive  upon  his 
investment  ?  Hffo- 


SECTION  XI, 


481 1  A  ^romiss07y  JVote  is  a  written  promise  to  pay 
a  certain  sum  of  money,  at  a  specified  time,  for  value 
received. 

COMMON   FORM   OF   A   PROMISSORY  NOTE. 
■/ol  'V-a/ae  -lecemec/.  c/'Veniy.     o/eioaic/. 


270  PERCENTAGE. 

482.  The  J^faker  of  a  promissory  note  is  the  person 
who  makes  or  signs  the  note  ;  and 

483.  The  ^ayee  is  the  person  to  whom,  or  to  whose 
order,  the  note  is  to  be  paid. 

481.  An  Jndorser  is  a  person  who  signs  his  name  upon 
the  back  of  a  note,  as  security  for  its  payment. 

485.  A  J^'^egoHable  A^ole  is  one  that  may  be  bought 
or  sold. 

Notes. — ^1.  A  note  payable  to  the  heather,  or  to  A.  B.  or  hearer,  is  negotiable 
without  indorsement.  Other  notes,  payable  to  A  B.  or  order,  or  to  the  order 
of  A.  B.,  are  not  negotiable  without  the  indorsement  of  A.  B. 

2.  The  sum  for  which  a  note  is  given  should  be  written  in  words  in  the 
body  of  the  note.     The  note  should  also  contain  the  words  "  value  received.^'' 

3.  If  no  mention  of  interest  is  made  in  a  note,  it  draws  interest  from  the 
day  it  is  due. 

486.  ^ays  of  Grace  are  three  days  allowed  after  the 
time  specified  in  a  note  has  expired,  before  the  note  is 
legally  due. 

487.  The  J)falurity  of  a  note  is  the  termination  of  the 
period  of  time  it  has  to  run.     It  is  the  last  day  of  grace. 

Note. — If  the  third  day  of  grace  falls  on  Sunday  or  a  legal  holiday,  the 
note  matures  on  the  second  day  of  grace. 

488.  A  Protest  is  a  written  notice,  in  due  form,  to  the 
indorser  of  a  note,  that  the  note  has  been  presented  to  the 
maker,  at  maturity,  for  payment,  and  has  not  been  paid  by 
him  ;  and  that  the  holder  looks  to  the  indorser  for  payment 
of  the  same. 

Notes.— 1.  A  protest  must  be  served  on  an  indorser  of  a  note  on  the  last 
day  of  grace,  to  hold  him  responsible. 

2.  Protests  are  usually  made  out  and  served,  by  an  officer  called  a  Notary 
Public. 

489.  A  !Sa7ik  is  an  institution  which  receives  deposits, 
loans  money,  and  issues  drafts,  bills  of  exchange,  and  bank 
bills  that  circulate  as  money. 

490.  A  Savi7igs-!Ba7lk  only  receives  money  on  dejposit, 
paying  interest  on  the  sums  deposited. 


BANKING.  271 

491.  A  ^ank-JV^oie  is  a  note  payable  at  a  bank.  Bank 
bills  are  also  called  bank-notes.     (See  214.) 

COIIMON  FORM   OF   A   BANK-NOTE. 
^/,000.  ^-^^     foi/,    o/an.    /^    /cf^ 

Gj,ia    <yf    f^ien/t'ce. 

492.  !Sank  ^isc0U7it  is  interest  paid  in  advance,  to  a 
bank,  for  the  loan  of  money  on  a  note. 

493.  The  J^ace  of  a  note  is  the  sum  due  at  maturity. 
If  the  note  is  on  interest,  the  face  is  the  amount  of  principal 
and  interest. 

494.  The  "Proceeds  of  a  note  is  the  face  of  the  note 
minus  the  interest. 

COM:i>tJTA.TIOI^S    IN"    BA-ISTKINGf-, 

495.  In  computing  bank  discount, 

The  face  of  the  note  to  be  discounted  is  the  principal,  or 
base  ; 

The  interest  on  the  face  of  the  note  for  the  given  time,  at 
the  given  rate,  is  the  bank  discount,  or  percentage  ; 

The  proceeds  of  the  note  is  the  difi'erence  ;  and 

The  product  of  the  rate  %  per'  annum  and  the  time  ex- 
pressed in  years,  is  the  rate.     Hence, 

I.  Face  of  note,  rate  and  \        {      ^^  ,    . 
time  given,   to  find    banki  is  \      Principal  raie,  and  time 
discount,                                )        I  g^ven,  to  find  interest. 

II.  Face  of  note,  rate  and  )  •  j  Base  and  rate  given,  to 
time  given,  to  find  proceeds,  )         \  find  difference, 

III.  Proceeds,   rate    and  \        (       ^  .  ^ 

lime  given,  to  find  fa^e  of  \  is  \     Differenceand  rate  given, 
note,  S       Vofindhase. 


272  PERCENTAGE. 

I'JtOBJjEMS. 

1.  "What  is  the  bank  discount  upon  a  note  for  $3,500,  due  in  4 
months,  at  6;^  ?  $51.25. 

2.  What  are  the  proceeds  of  a  note  for  $650,  due  in  90  days,  if 
discounted  at  a  N.  Y.  bank  ?  $638.25. 

3.  A  note  for  $7,350,  due  Aug.  12,  18G8,  was  discounted  at  a 
bank  in  Charleston,  S.  C,  May  20, 1868.   What  were  the  proceeds  ? 

4.  What  sum  can  be  realized  at  a  bank,  upon  a  note  for  $11,500, 
due  in  30  days,  at  Qfo  discount  ? 

5.  A  bank  loans  $4,500  on  a  note  payable  in  4  months,  discount- 
ing it  at  %fo.    What  is  the  face  of  the  note  ?  $Jf,  62646. 

6.  I  obtained  $237,  at  a  bank  m  New  Orleans,  on  a  note  due  in 
8  months.    What  was  the  face  of  the  note  ? 

7.  A  merchant  wishes  to  borrow  $2,000  at  a  bank.  For  what 
amount  must  he  make  his  note,  due  in  60  days,  if  he  gets  it  dis- 
counted at  Qfo  ? 

8.  What  will  be  the  proceeds  of  a  note  for  $4,500,  due  in  6 
months,  discounted  at  a  bank  in  San  Francisco  ?  $4^271.25. 

9.  A  manufacturer  in  St.  Louis  wishes  to  borrow  $2,000  with 
which  to  pay  his  men.  If  he  makes  a  bank-note,  due  in  30  days, 
what  will  be  the  face  of  the  note  ?  $2,011.06. 

10.  A  merchant  buys  a  bill  of  goods  in  Milwaukee,  amounting  to 
$3,700,  on  3  months'  time,  or  5^  off  for  cash.  If  he  borrows  the 
money  at  a  Milwaukee  bank,  will  he  make  or  lose,  and  how  much  ? 

11.  I  had  a  note  for  $460,  dated  Detroit,  Jan.  9, 1869,  and  due  in 
8  months,  with  interest.  March  19,  1869,  I  had  it  discounted  at  a 
bank,  at  8^C     How  much  did  I  realize  from  the  note  ? 

12.  For  what  sum  must  a  bank-note,  due  in  5  months,  be  made, 
to  have  it  produce  $1,856  when  discounted  at  the  First  National 
Bank  of  Boston  ?  $1, 904.57. 

13.  Find  the  face  of  a  note,  due  in  G  months,  on  which  I  can 
borrow  $875  at  a  Chicago  bank.  $921.86. 

14.  In  payment  of  a  debt,  I  took  Chas.  Marshall's  note  for 
$1,600,  payable  at  the  Sixth  National  Bank  of  Philadelphia,  in  6 
months,  with  interest.  Four  months  afterward,  I  had  it  dis- 
counted at  the  First  National  Bank  of  Harrisburg.  What  were  the 
proceeds  ? 


EXCHANGE.  273 

SECTION  XII. 

B  X  C  H  A  JV'  G  JE  , 

496.  .Exchange  is  a  commercial  transaction,  in  which 
a  party  in  one  place  pays  money  to  a  second  party  in 
another  place,  by  means  of  an  order  upon  a  third  party,  and 
without  the  transmission  of  money. 

497.  A  ^7^aft,  or  :Bill  of  Uxc?ia7ige,  is  a  written  or- 
der for  money,  drawn  in  one  place  and  payable  in  another. 

Example.^ — A  Chicago  merchant,  wishing  to  pay  a  debt  in 
New  York,  buys  at  a  bank  in  Chicago,  a  draft  on  a  New  York 
bank,  payable  to  the  order  of  the  party  in  New  York.  The 
Chicago  merchant  sends  this  draft  to  his  creditor  in  New 
York,  and  the  latter  indorses  it,  presents  it  to  the  New 
York  bank,  and  receives  the  face  of  the  draft  in  money. 

Notes.— 1.  Any  party  may  give  a  draft,  or  "  draw  "  on  another  party,  if 
the  second  party  is  debtor  to  the  first. 

2.  A  draft  payable  in  the  same  country  in  which  it  is  drawn,  is  an  Inland 
Bill  of  Exchange ;  and  one  drawn  in  one  country,  and  payable  in  another,  is 
a  Foreign  Bill  of  ExcJiange. 

498.  A  SigJit  ^7*aft  is  a  draft  payable  "  at  sight,"  L  c, 
when  it  is  presented  ;  and 

499.  A  2tme  ^raft  is  a  draft  payable  at  a  future  time 
named  in  it. 

Note. — Grace  is  allowed  on  time  drafts,  but  not  on  sight  drafts. 

500.  There  are  usually  four  parties  to  a  transaction  in 
exchange,  viz.. 

The  H) rawer  or  Maker  of  the  draft ; 

The  ^uyer  or  ^emiitery  or  the  party  who  purchases 
the  draft ; 

The  ^7'aweey  or  the  party  on  whom  the  draft  is  drawn  ; 

The  "Pa/yee,  or  the  party  to  whose  order  the  draft  is 
made  payable. 

Note.— The  maker  and  remitter  of  a  draft  may  be  the  same  party,  in 
which  case  there  will  be  but  three  parties  to  the  transaction. 
1£* 


274  PERCENTAGE. 

COMMON   FORM   OF  DSAFT. 

STAMP.  I  c^«  c^aj^  a/^Ai  ^^/^ pay  to  tha  order  of 

^ai/iei    ^  Mlo^^eid ■■■■■ 

/'/eejf   /func/uc/ Dollars, 

Value  received,  and  charge  to  account  of 

M^     ^oi/  ^/^. 

Note.— The  words  At  SigU,  in  place  of  "  Ten  days  after  sight "  would 
make  the  above  a  sight  draft. 

COIMFXIT^TIONS    IIS-    EXCH^^STGE. 

501.  Drafts  or  bills  of  exchange  are  bought  at  par,  at 
a  premium,  or  at  a  discount. 

The  face  of  the  draft  is  the  base  ; 

The  rate  %  of  exchange  is  the  rate  ; 

The  premium  or  discount  is  the  percentage  ;  and 

The  proceeds  of  the  draft  is  the  amount  or  difference. 
Hence, 

I.  Face  of  draft  and  rate  )         c 

%  of  exchange  gimn,  to  fined  is  \  ^^"^^  ""'^  rate  given,   to 
premium  or  discount,  )        (  find  percentage. 

II.  Proceeds  and  rate  %\         i       ,  -..^ 

r,       7  .         i     ^  j\    •     \      Amount     or     difference 

of  exchange  muen,  to  jina>  is  -<       ,  .  f  -, -, 

r        jy  1  ^  f,  \         I  ana  rate  given,  to  find  base. 

Notes. — ^1.  The  face  of  a  draft  plus  the  premium,  or  minus  the  discount, 
is  the  proceeds. 

3.  The  subject  of  Foreign  Exchange  is  not  considered  in  this  book.  A 
full  presentation  of  it  will  be  found  in  the  Academic  Arithmetic  of  this 
Series. 

PU  OBLEMS. 

1.  At  2><fo  premium,  how  much  will  it  cost  me  to  remit  $2,700  by 
draft  from  Grand  Rapids,  Micb.,  to  Philadelphia  ? 

3.  How  much  will  it  cost  me  to  make  a  remittance  of  $280  from 
New  York  to  Baltimore,  exchange  on  Baltimore  being  at  ^(fo  dis- 
count ?  $278.60. 


AVERAGE    OF    PAYMENTS.  275 

3.  If  I  buy  a  draft  for  $1,285  in  Pittsburg,  to  send  to  Louisville, 
at  l^  discount,  how  much  does  it  cost  me  ?  $1,278.58. 

4.  How  much  will  a  man  in  Galena  have  to  pay  for  a  draft  on 
Boston  for  $532,  exchange  being  at  3|^  premium  ? 

5.  A  man  at  Springfield,  111.,  wishing  to  remit  to  a  creditor  in 
Harrisburg,  Penn.,  buys  a  draft,  at  2|-^  premium,  and  pays  $240 
for  it.     What  is  the  face  of  the  draft  ?  $2J!tO. 

6.  A  man  in  Brooklyn,  having  $145.50  belonging  to  a  man  in  St. 
Paul,  Minn.,  purchases  a  draft  with  it  on  a  St.  Paul  banker,  at  3^ 
discount.     What  is  the  face  of  the  draft  ? 

7.  What  must  I  pay  in  Wheeling,  W.  Va.,  for  a  draft  on  N.  Y.  for 
$1,200,  payable  30  days  after  sight,  exchange  being  at  If^  premium? 

Note. — On  a  time  draft,  both  the  discount  and  the  rate  of  exchange 
must  be  computed  on  the  face  of  the  draft.  $l,2lJ^.Jj,0. 

8.  A  St.  Louis  banker  discounts  a  draft  on  Baltimore  for  $860, 
payable  90  days  after  sight,  exchange  on  Baltimore  being  at  4|^ 
premium.     What  does  he  pay  for  the  draft  ?  $885.31. 

9.  A  grocer  in  Rock  Island  paid  $611.70  for  a  draft  on  New 
York,  payable  60  days  after  date,  when  exchange  on  New  York 
was  at  3^  premium.     What  was  the  face  of  the  draft  ? 

10.  A  broker  in  Columbus,  Ohio,  pays  $352.63  for  a  draft,  pay- 
able at  Knoxville,  Tenn.,  30  days  after  sight,  at  \\(fo  discount. 
What  is  the  face  of  the  draft  ?  $SQO. 


SECTION   XIII. 

502.  Average  of  ^aymenls  is  the  process  of  finding 
the  time  for  paying  several  sums  due  at  different  times,  so 
that  no  loss  shall  be  sustained  by  either  party. 

503.  The  Term  of  Credit  is  the  time  during  which  no 
interest  is  paid. 

5®4.  The  Average  2'erm  of  Credit  is  the  average 
time  during  which  no  interest  is  paid  on  different  debts  due 
at  different  times. 


276  PERCENTAGE. 

505.  The  ^qiiated  Thne  is  tlie  date  on  whicii  several 
debts  due  at  different  times  may  all  be  paid,  v/itliout  loss  to 
either  creditor  or  debtor. 

5C6.  Averagi7i(/  an  Account  is  the  process  of  find- 
ing the  mean  or  equated  time  for  the  payment  of  the 
balance  of  the  account. 

Equation  or  average  of  payments  is  of  two  kinds,  Simple 
and  Compound. 

507.  Simpte  Average  is  one  in  which  the  sums  are 
cither  all  debits  or  all  credits. 

508.  Compound  Avet^age  is  one  in  which  some  of  the 
sums  are  debits,  and  some  are  credits. 

509.  A  JF^ocal'Date  is  any  date  taken  as  a  standard  of 
reference,  and  with  v/hich  each  given  date  is  compared. 

Tho  terms  of  Credit  beginning  at  the  same  Date. 

510.  E?.  July  1,  I  purchase  a  saw-mill  for  $1,200,  paying 
1400  down,  and  agreeing  to  pay  $200  in  3  months,  $400  in  6 
months,  and  $200  in  9  months,  without  interest.  What  is 
the  average  time  for  the  payment  of  the  whole  amount  of 
the  purchase  money? 

Explanation.  —  solution. 

The  cash  payment  ^^  ^  ^  ?,  ^^'''  r^''^''''-  ^    ^  ^  ^  ^  , 

p    &>Ac^c^\:  200for3mo.=  S   GOOforlmo. 

of    $400     has   no  100    "   6    "  =    2A00   "  1    " 

term  of  credit,  and  2  00   "   9    "  =    18  00   '^  1    " 

only  forms  a  part  ~^  12OO    "    ?    "  =$J/.800  "  1    « 
of    the    purchase 

money.     The   use  $Ji.800  [ $1200 

of      $200        for      3  !{-'=  Jf-  mo.,  average  term  of  credit. 

months      is     the  j^iy  i+^  rno.  =  Nov.  1. 

same  as  that  of  3 

times  $200,  or  $600,  for  1  month  :  the  use  of  $400  for  6 
months  is  the  same  as  that  of  $2,400  for  1  month  ;  and  tho 
use  of  $200  for  9  manths  is  the  same  as  that  of  $1,800  for 


AVERAGE    OF    PAYMENTS.  277 

1  month.  The  sum  of  the  payments,  due  at  different  times, 
is  $1,200,  and  of  the  equivalent  sums  for  1  month  is  $4,800. 
The  use  of  $4,800  for  1  month  is  the  same  as  the  use  of 
$1,200  for  as  many  months  as  the  number  of  times  $1,200 
are  contained  in  $4,800.  Dividing  $4,800  by  $1,200,  we  ob- 
tain 4—4  months,  the  average  term  of  credit ;  and  July 
1  +  4  months  =  Nov.  1,  the  time  required.     Hence, 

TJie  product  of  the  sum  of  all  the  payments  multiplied  by 
the  average  term  of  credit,  equals  the  sum  of  the  products  of  all 
the  payments  multiplied  by  their  respective  terms  of  credit. 

511.  On  this  principle  is  based  the 

(Hiite  for  finding  the  a%'erage  time  ofpaymetit,  7i'hen  t/ie 
ter77is  of  credit  all  begin  at  the  same  date, 

I.  Multiply  each  payment  by  its  term  of  credit,  and  add  all 
the  payments,  and  also  all  the  products. 

II.  For  the  average  term  of  credit,  divide  the  sum  of  the 
products  by  the  sum  of  the  paijments. 

III.  For  the  average  time  of  payment,  add  the  term  of  credit 
to  the  given  date. 

Tit  OBL  EMS, 

1.  March  8,  I  bought  a  building  lot  for  $800,  paying  $200  down, 
and  agreeing  to  pay  $200  in  4  months,  $200  in  8  months,  and  $200 
in  12  months.  Had  I  given  my  note  for  the  payment  of  the  whole 
amount  at  once,  at  what  date  should  it  have  been  made  payable  ? 

Sept.  8. 

2.  May  29,  a  merchant  bought  bills  of  goods  as  follows:  $825  on 
a  credit  of  3  months,  $675  on  4  months,  $450  on  2  months,  and 
$800  on  1  month.  What  is  the  average  time  for  the  payment  of  the 
whole  amount  ?  Aug.  I4. 

3.  On  the  first  day  of  May,  D  hired  a  house  at  $300  per  annum, 
agreeing  to  pay  the  rent  quarterly.  What  would  be  the  equated 
time  for  the  payment  of  the  whole  ?  Dec.  16. 

4.  To-day  I  owe  $150  due  in  30  days,  $200  due  in  60  days,  and 
$250  due  in  90  days.  If  I  give  my  note  for  the  whole  amount, 
made  payable  at  the  average  time,  when  vail  the  note  be  due?_^^ 


278  PERCENTAGE. 

5.  What  is  the  average  time  for  the  payment  of  5  notes,  all 
bearing  date  June  17 ;  one  for  $300  due  in  3  months,  one  for  $500 
due  in  5  months,  one  for  $150  due  in  7  months,  one  for  $350  due 
in  9  months,  and  one  for  $200  due  in  1  year  ? 

Cj^SE    II. 
The  terms  of  Credit  beginning  at  different  Dates. 

512.  Ex.  March  21, 1  bought  a  horse  for  $175  on  a  credit 
of  4  months  ;  June  5,  a  harness  for  $55  on  3  months,  and 
a  top  carriage  for  $225  on  4  months.  What  is  the  average 
time  for  the  payment  of  the  three  debts  ? 

Explanation. —  solution. 

To  find  the  dates  ^^'^'  ^i-\-J^rno.  =  July  2 1 

1  .  ■.    ,  1  June     6  +  3   "    =  Sept.     5 

on  which  the  sev-  ,,        ^|;^   ,,    =  Oct,       5 

era]  payments  fall 

due,   we   add   to      $  17  5  cash,  July  21. 
each  date  the  term  5  5  for  J^6  da.  =  $    253  0  for  1  da. 

225    "     76  "    =     17 100   "    1  " 


of  credit.     Since 
the  first  payment      SJ^55   "      'i    "    =  $  196 30   ''    1  '' 
falls  due  July  21,      ^  19  630  \  $455 
we  take  that  date      _^1^  I       J,3=:J,3da.{  ""'^JTelr' 
as    a  focal  date.  lJf30 

Comparing  this  13_65_ 

focal     date    with  65  =/jV,  less  than  |  da.,  dropped. 

the  dates  on  which  July  21  +  J^3  da.  =  Sept.  2. 

the   several  pay- 
ments mature,  we  have  $175  due  at  the  focal  date,  $55  in 
46  dsijs,  and  $225  in  76  days.     Proceeding  as  in  Case  I., 
we  find  the  average  term  of  credit  to  be  43  days  after  July 
21,  or  Sept.  2,  the  average  time  required.     Hence, 

When  the  terms  of  Credit  begin  on  different  dates, 
The  earliest  date  on  which  any  one  of  the  payments  matures, 
may  be  taken  as  the  focal  date  ;  and  the  time  between  this  date 
and  the  date  on  which  each  of  the  other  payments  matures,  may 
be  taken  as  the  term  of  credit  of  that  payment. 


AVERAGE    OF    PAYMENTS.  279 

513.  On  this  principle  is  based  the 

^iile  fo7'  fiiidlng  the  average  time  ofpaj'77?e7it,  when  the 
tei^ms  of  credit  begin  at  different  dates. 

I.  Find  the  time  on  which  each  payment  becomes  due,  by 
adding  the  term  of  credit  to  the  date  of  the  transaction. 

II.  For  the  focal  date,  take  the  earliest  dale  on  which  any 
one  of  the  payments  matures  ;  and  for  the  term  of  credit  of  each 
payment,  take  the  time  between  the  focal  date  and  the  time  on 
which  the  payment  matures. 

m.  For  the  average  time  of  payment,  find  the  average  term 
of  credit,  and  add  it  to  the  focal  date. 

PR  OBLEMS. 

6.  Find  the  average  time  for  the  payment  of  the  following  bills: 

Jan.  10,  $415  on  3  months ; 
Feb.  25,    175    "    4       " 
Apr.    5,    350   "    3      " 

35  days  from  Apr.  10,  or  May  15. 

7.  What  is  the  equated  time  for  the  payment  of  three  notes,  one 
for  $650,  dated  July  13,  and  due  in  90  days ;  one  for  $555,  dated 
July  35,  and  due  in  60  days ;  and  one  for  $445,  dated  Aug.  14,  and 
due  in  30  days  ?  Sept.  27. 

8.  George  Adams  bought  provisions  as  follows : 

Mar.  14,    40  bar.  beef,   ®  $17.50,  at  3  months. 
May     1,     60     "     pork,  @    34  " 

June  10,  150     "    flour,  @      8        Terms  cash. 
What  is  the  average  time  for  the  payment  of  the  whole  ? 

9.  I  make  the  following  advances  of  money  for  a  friend :  May 
19,  $107;  May  38,  $35  ;  June  37,  $130;  July  3,  $70;  Aug.  34,  $80, 
and  Sept.  11,  $175.  If  I  take  his  note  for  the  whole  amount,  dated 
at  the  equated  time,  what  will  be  the  date  of  the  note  ?    July  19. 

10.  B  works  for  C  6  months  from  May  15,  at  $60  per  month, 
his  wages  to  be  paid  one  half  monthly,  and  the  other  half  in  3 
months.  In  lieu  of  receiving  his  pay  according  to  contract,  he 
takes  C's  note  for  the  whole  amount,  bearing  date  from  the  average 
time,  with  interest.     What  is  the  date  of  the  note  ?  Oct.  15. 


280 


PERCENTAGE. 


C^SK   III. 
Accounts  containing  both  Debits  and  Credits. 
5!li  Computations  in  compound  average  are  based  upon 
the  following  equitable  principles  : 

I.  All  payments  made  before  the  average  term  of  credit  ex- 
pires, should  draw  interest  from  the  time  they  are  made  ;  and 

II.  All  debits  not  paid  till  after  the  average  term  of  credit 
expires,  shoidd  draw  interest  from  the  expiration  of  the  aver- 
age term  of  -credit. 

515.  Ex.  Find  the  average  time  for  the  payment  of  the 
balance  of  the  following  %.  : 

Dr.  KoBEKT   Lansing.  Cr. 


1869 

1869 

i 

Jan. 

4 

To  Mdse.@4  mo. 

325 

50 

Feb. 

15 

By  Cash, 

200 

00 

u 

16 

"     do.      4    " 

37 

50 

Apr. 

17 

"      do. 

75 

00 

Mar. 

1 

"     do.      4    " 

162 

50 

u 

26 

"    do.     4    " 

250 

00 

EXPLANATION.- 

Ist,  AVe  find  the 
average  time  of 
the  debits  to  be 
June  12  ;  and, 
2d,  The  aver- 
age time  of  the 
credits  to  be 
Mar.  4 ;  both  by 
Case  II. 

3d,  Since  $275 
was  paid  100 
days  before  it 
was  due  {i.e.  be- 
fore the  average 
time  at  Vv^hich 
the  debits  were 


SOLUTION. 

1st.— Averaging  the  Debits, 

Jan.     Jf.  +  Jfmo.  =  May    J,.. 

"     16 -{-J^   "     =    "      16. 

Mar.    i  +  4    "     =  July    1. 

"     26  +  Jf   "     =    "      26. 

Focal  date,  May  4-. 

So  2  5.5 0  cash  payment.  May  J^. 

3  7.50  for  12  da.  =  $      1^50  for  1  da. 


162.50 

250 


$775.50   " 

$30625.0 

23265 

~~73600 
69795 


58 
83 


9Jf25 
=      20750 

=  $30625 


$7  75.5 
39.4.  —  39  da.i^^^'^''^^^.*^''"'' 


3805^0 
31020 
~70Jd 
May  4  +  39  da. 


0/ credit. 


J-  ^Q^  Average  ti77ie /or 

June  i-v  j  payment  of  d'ehita. 


AVERAGE    OF    PAYMENTS. 


281 


due),    interest 
must  be  allowed 
on  $775.50- 
$275  =  $500.50, 
the  balance,  for 
100  days   (511, 
I.);  or,  whicli  is 
the  same  thing, 
the  term  of  cred- 
it for   $500.50, 
the    balance, 
must  be  extend- 
ed a  sufficient 
time  to  average 
the  use  of  the 
$275  paid    100 
days    before   it 
was  due.      We 
find  this   addi- 
tional   term   of 
credit    for    the 
balance  of   the 
account,  by  mul- 
tiplying    $275, 
the  sum  of  the 
credits,  by  100, 
the   number  of 
days  before  the 
maturity  of  the 
debits,  and  di- 
viding $27,500, 
the  product,  by 
J55500.50,  the  bal- 
ance of  the  %., 
as  in  511.     The 


2(1.— Averaging  the  Credits. 

Focal  date,  Feb.  15. 

$200  cash  payment,  Feb.  15. 

T5^for  Gl  da.  =  S 45 7 5 for  1  da. 

$215  "     ?    " 


$4515 


$4575 
275 


\$275 


1825 
1650 

~'rwo 

1650 
100 


-I  o  a         -iiy  J    S  ■A'Berage  term 
I  0.0  —  1/  aa.}     of  payments. 


Feb.  15  +  17  da.  =  Mar.  4\^l}'^'4mZs. 


Sd. — Averaging  Debits  and  Credits. 

Focal  date,  June  12. 

From  Mar.  4  to  June  12  =  100  da. 

$775.50  -  $275  -  $500.50. 
$275     for  100  da.  =  $27500 for  1  da. 
500.50''     ?      "    =    27500   "   1  " 


$27500.0  \  $500.5 
25025 


24750 
20020 

4730^^0 
45045 

2255 

June  12  +  55  da. 


■  (  Average  time 

54.9  =  55  da.}  o/lal.  o/c^c. 


Aug.  ei^f/^fp'^y""'''* 

if        \ofbal.ofal^ 


282 


PERCENTAGE. 


result,  55  days,  added  to  June  12,  the  date  of  the  maturity 
of  the  debits,  gives  Aug.  6,  average  time  required. 

Had  the  date  of  the  average  term  of  the  credits  been 
later  than  that  of  the  average  term  of  the  debits,  i.  e.,  after 
June  12,  we  should  have  dated  back,  or  subtracted,  the 
average  time  of  the  bal.  of  the  %.  from  June  12.     Hence, 

516,  ^if^e  fo7^  Co77iputmg  Compound  Average, 

I.  Find  the  average  term  of  the  debits  and  credits  separately. 
n.  For  the  average  term  of  credit  of  the  halance  of  the 

account^  tahe  the  average  date  of  the  larger  of  the  two  sides 
of  the  account  for  a  focal  date,  multiply  the  smaller  side  hy 
the  difference  in  time  between  its  date  and  this  focal  date,  and 
divide  the  product  hy  the  balance  of  the  account. 

HI.  For  the  date  of  the  average  time,  count  foriuard  from 
the  last  focal  date,  if  the  larger  side  of  the  account  falls  due 
later;  and  bachivard,  if  it  falls  due  earlier, 

PJtOJiLE3IS. 

II.  Balance  the  following  %.,  and  find  when  it  is  due  : 


Dr. 

E. 

M.  D 

ANIELS    &    Co. 

Cr. 

1869 

1869 

June 

14 

To  Mdse.  ®  3  mo. 

450 

00 

Sept. 

3 

By  Cash, 

400 

00 

Aug. 

25 

u        u            g    a 

175 

00 

Nov. 

2 

"      do. 

150 

00 

Oct. 

11 

u        a            Q    u 

425 

00 

u 

30 

"      do. 

225 

00 

Fel.  17,  1870. 
12.  What  is  the  balance  of  the  following  %.,  and  when  is  it  due  ? 
Dr.  John  G.  Andeeson.  Cr. 


1869 

1869 

Oct. 

9  To  Mdse.®  3  mo. 

300 

00 

Nov. 

24 

By  Cash, 

25 

00 

Nov. 

18    "     do.       3    "  ; 

329 

00 

Dec. 

4 

"      do. 

500  00 

u 

27 

"     do.      3    '' 

142 

00 

a 

30 

"   Note, 

150  00 

Dec. 

19 

"     do.       3    " 

256 

00 

$352;  June  25,  1870. 


REVIEW     PROBLEMS. 


283 


13.  If  a  note,  drawing  interest,  be  given  to  balance  tlie  following 
%.,  for  what  sum  will  it  be  drawn,  and  What  will  be  its  date  ? 


Dr. 


Ames  &  Potteb. 


Gr, 


1869 

1869 

Mar. 

17 

ToMdse.(g^2mo. 

325 

00 

July 

25 

By  Cash, 

125 

00 

Apr. 

20 

"     do.      3    " 

108 

00 

Aug. 

17 

"     do. 

300 

00 

July 

18 

"    do.       Cash, 

264 

00 

Oct. 

24 

Draft  on  KY., 

350 

00 

Aug. 

11 

"     do.  ®4mo. 

50 

00 

j 

Sept. 

35 

•'    do.      3    " 

125 

00 

$97;  May  25,  1868, 


SECTION  XIV. 
it^riByr  tiioszbms  ij\r  !Pjej^cbjvta.gb. 

1.  What  is  13|^  of  837  bushels  of  wheat  ?  1U.39  lu. 

2.  If  I  sell  a  sewing-machine  for  $50  that  cost  me  $56,  what  fo  do 
I  lose  ?  lOff,. 

3.  How  much  will  100  shares  of  N.  Y.  C.  R.R.  stock  cost  me,  at 
154|^,  brokerage  Ifo  ? 

4.  Of  every  1,375  persons  25  years  old,  1,265  will  live  to  the  age 
of  26.    What  fo  of  persons  25  years  old  die  annually  ?  8^. 

5.  A  real  estate  agent  receives  $35  for  selling  a  farm,  his  rates  of 
commission  being  1%  from  the  buyer,  and  1^^  from  the  seller. 
What  is  the  price  obtained  for  the  farm?         .  $1,400. 

6.  A  note  for  $36.50,  dated  June  27  of  last  year,  was  paid  April 
4  of  this  year,  with  interest  at  10^^.     What  was  the  amount  paid? 

7.  A  commission-merchant  receives  $820  with  v/hich  to  buy 
goods,  after  deducting  his  commission  of  2-l-fo.  How  much  does  he 
expend  for  goods  ?  $800. 

8.  I  borrow  $4,000  on  my  note  in  Portland,  Me.,  Feb.  21,  and 
loan  the  money  in  Syracuse,  N.  Y.,  Feb.  23.  If  the  money  is  paid 
to  me,  Nov.  12,  and  I  pay  my  note  Nov.  13,  how  much  do  I  gain  ? 

$26.78. 


284  PERCENTAGE. 

9.  Three  men  own  a  mill.  C's  share,  which  is  $2,550,  is  60^  of 
B's,  and  B's  share  is  85^  of  A's.     What  is  the  value  of  the  mill  ? 

$11,800. 

10.  A  merchant  paid  a  premium  of  $363.12|  for  a  policy  of  in- 
surance covering  $8,000  on  his  store,  and  $13,750  on  his  goods. 
What  was  the  rate  ?  1^%. 

11.  If  a  clergyman's  salary  is  $1,500,  and  he  pays  $175  for  house 
rent,  what  is  his  income  tax  ? 

13.  If  the  annual  rate  on  a  life  at  50  years  is  $4,439  per  $100, 
payable  semi-annually,  what  will  be  each  payment  on  a  policy  for 
$3,500  ? 

13.  What  is  the  premium,  at  |^,  for  insuring  a  farm-house  for 
$800,  a  barn  for  $750,  and  the  hay  and  grain  for  $1,200  ? 

14.  One  morning,  five  canal  boats  were  weighed  at  the  weigh- 
lock  in  Utica.  The  cargo  of  the  first  weighed  64  tons,  and  the 
cargoes  of  the  others  weighed,  respectively,  85^,  67|^,^,  130^^,  and 
56^^  of  that  amount.  What  was  the  total  weight  of  the  five 
cargoes  ?  27J^.Jf  tons. 

15.  I  buy  a  draft  in  Portsmouth,  N.  H.,  for  $350,  payable  in 
Providence,  R.  I.,  60  days  after  sight,  exchange  |^  premium.  How 
much  does  the  draft  cost  me  ?  $2JfS.62t. 

16.  In  a  village  school  of  four  departments,  33  pupils  are  in  the 
first  department,  44  in  the  second,  54  in  the  third,  and  60  in  the 
fourth.     What  ^  of  all  the  pupils  attend  each  department  ? 

17.  A  note  for  $360,  drawing  %fo  interest,  was  dated  April  10, 
1868,  and  $335  was  paid  on  it,  Jan.  19,  1869.  What  amount  was 
due  Nov.  3,  1869  ? 

18.  A  stock  jobber  bought  100  shares  of  Pacific  R.R.  stock,  at 
117.  He  sold  55  shares  at  104|^,  and  exchanged  the  balance  at  108 
for  Ocean  Bank  stock  at  135.  He  afterward  sold  the  bank-stock 
at  1441-.     Did  he  gain  or  lose  ?  He  lost  $750.50. 

19.  A  Va.  planter  took  up  a  note  for  $843,  Oct.  31,  1869,  that 
was  dated  May  29,  1867.     What  interest  had  accrued  ?    $122.52. 

30.  A  box  of  soap,  marked  60  lb.,  loses  l^fo  by  drying.  What  is 
its  actual  weight  ? 

31.  If  a  man  buys  U.  S.  5-30  bonds  at  106|-,  and  sells  the  gold 
interest  at  137|-,  what  fc  in  currency  does  his  investment  pay  him  ? 


KEVIEW    PROBLEMS.  285 

22.  A  fruit  dealer  bought  quinces  at  $1.60  per  bushel.  After 
assorting  them,  he  sold  the  best  at  35^  profit,  and  the  others  at  15^ 
loss.     What  were  his  selling  prices  ? 

23.  In  building  a  house  which  cost  $1,480,  43^  of  the  cost  was 
for  labor,  and  the  balance  for  materials.  What  was  the  cost  of  the 
materials  ?  $8JiS.60. 

24.  How  shall  a  merchant  mark  carpeting  that  cost  him  $1.42 
per  yard,  so  that  he  can  fall  8^  from  the  marked  price,  and  still 
make  25^  ?  At  $1.68^}  per  yd. 

25.  A  mechanic  buys  a  city  lot  for  $600,  payments  $250  cash, 
and  the  balance  in  1  year  without  interest.  In  6  months  he  pays 
the  balance,  less  the  discount,  at  Q^.     How  much  does  he  pay  ? 

26.  A  field  of  11  acres  yielded  16.5  bushels  of  wheat  to  the  acre ; 
the  cost  of  seed  and  labor  was  $193.60,  and  the  wheat  brought 
$1.60  per  bushel.     What  fo  profit  did  the  crop  pay  ?  50fc. 

27.  What  is  the  face  of  a  sight  draft  that  costs  $360,  exchange 
being  at  1^-%  premium  ? 

28.  An  Oswego  miller  buys  a  draft  for  $2,500  on  Chicago,  at  |^ 
discount.  He  remits  the  draft  to  a  grain  buyer  in  Chicago,  with 
instructions  to  invest  the  proceeds,  less  his  commission  of  1^,  in 
wheat.  Is  the  miller's  gain  on  the  exchange  more  or  less  than 
the  grain  buyer's  commission  ?  $6. 

29.  What  is  the  face  of  a  note  due  in  2  mo.  15  da.,  the  pro- 
ceeds of  which,  discounted  at  the  Firet  National  Bank  of  Burling- 
ton, Vt.,  are  $370.12^  ?  $375. 

30.  A  drover  paid  $4,325  for  cattle,  and  $1,498  for  marketing 
them,  and  they  sold  for  $6,375  at  60  days.  What  were  his  net  cash 
profits,  money  being  worth  8fc'i  $Ji,68.12. 

31.  Memorandum  :— Face  of  mortgage,  April  23,  1866,  $3,275. 
Indorsements,— Sept.  4,  1866,  $845  ;   Feb.   27,   1868,  $150 ; 

Aug.  19,  1868,  $75 ;  Jan.  7,  1869,  $1,250. 
What  was  due  July  1,  1869,  interest  at 

32.  Find  the  time  from  which  interest 
should  be  reckoned  on  the  sum  of  the  debts 
in  the  margin,  if  all  of  them  are  paid  when 
the  last  one  is  due.  Aug.  1. 


?      i 

$1,417.92 

'. 

400  due  Mar. 

3. 

325 

„    May 

19. 

1,000 

»»      n 

25. 

625 

„    Sept. 

4. 

1,275 

„   Nov.  12. 

286  PERCENTAGE. 

83.  If  you  borrow  $620  for  1  yr.,  at  8^  interest,  and  5  mo.  afler* 
-ward  you  pay  $314,  how  much  will  you  owe  at  the  end  of  the  year? 

34.  A's  property  is  assessed  at  $6,750,  and  B's  at  $13,575,  and 
A's  tax  is  $52.65.     How  much  is  B's  tax  ?  $105.89. 

35.  A  liquor  dealer  imported  45  casks  of  wine,  invoiced  36  gal. 
each,  at  $1.50  per  gal.  He  paid  $1,75  per  cask  for  transportation, 
a  specific  duty  of  $1  per  gal.,  and  25^  ad  valorem  duty.  Deduct- 
ing the  customary  allowance  of  2^  for  leakage,  what  did  the  wine 
cost  him  ?  $4.,69L70. 

36.  How  many  shares  of  bank-stock,  at  par,  can  a  stock  broker 
buy  for  $4,522.50,  less  his  brokerage  of^fo  ? 

(37) 

(Dn-e  ^eal  a/'/ei  e/a'/a,    Q^/tiome^e  >/(> /tay  i^nt/ie^uji   ^.    ^^fnt^e^ 

Indorsements :  June  18,  1869,  $125 ;  Oct.  25,  1869,  $475. 
How  much  was  due  on  settlement.  Mar.  4,  1870  ? 

38.  When  gold  is  worth  135,  which  is  the  better  investment, 
U.  S.  10-40's  at  97i,  or  5-20's  at  107|  ? 

39.  Find  the  average  time  for  the  payment  of  the  following 

Statement  of  %. 
In  %  with  %  113.  Perkins  ^  OTo.,  ^^ 


/&^<S'. 

(^c-/.        /p,    c^  ^ifc^e.    @   S^o., 

/    S/S 

.J^.       ^/     ,,       c/a.          SO  e/a. 

/^/ 

^ec.    S<f,     ,,       c/a.               ^ad/. 

/.oso 

/<r^p. 

s^s 

A^^/ 

40.  I  have  a  mortgage,  bearing  interest  at  7^,  payable  annually. 
Will  I  gain  or  lose  by  selling  it  at  \(fo  discount,  and  investing  tho 
proceeds  in  U.  S.  6's  of  '81,  at  102|-,  when  gold  is  quoted  at  125  ? 


SECTION   I. 

2iA.  210, 

517t  ^atio  of  mimbers  is  the  relative  value  of  one  num- 
ber to  another  of  the  same  land. 

518.  The  jTerms  of  a  ratio  are  the  numbers  whose 
values  are  compared. 

519.  The  Antecede?it  is  the  first  term  ;  and 

520.  The  Consequent  is  the  second  term. 

Example. — The  relative  value,  or  the  ratio,  of  15  to  5,  is  3. 
In  this  example,  15  and  5  are  the  terms  of  the  ratio,  15  is 
the  antecedent,  and  5  is  the  consequent. 
Note. — The  terms  of  a  ratio  are  called  a  CoupleL 

521.  Baiio  is  obtained  by  dividing  the  antecedeiit  by  the  con- 
sequent. 

522*  The  Slf/n  of  ratio  is  the  colon  ( : ).  It  is  -^Titten 
between  the  terms  of  a  ratio,  and  is  read,  "  the  ratio  of." 
Thus,  15  :  5  is  read,  "  the  ratio  of  15  to  5." 

Since  the  ratio  is  obtained  by  dividing  the  antecedent  by 
the  consequent,  ratio  may  also  be  expressed  by  writing  the 
antecedent  for  the  numerator,  and  the  consequent  for  the 
denominator  of  a  fraction.     Thus,  15  :  5  —  -'/-. 

523.  A  Simple  !RaHo  is  the  ratio  of  any  one  number 
to  another  number  of  the  same  kind  ;  and 

521.  A  Compotcnd  UlaHo  is  the  ratio  of  the  products 
of  the  corresponding  terms  of  two  or  more  simple  ratios. 

Example. — The  ratio  of  15  to  5  is  a  simple  ratio ;  and  the 
ratio  of  the  products  of  the  corresponding  terms  of  the 
ratios  15  :  5  and  8  :  2,  or  15  x  8  :  5  x  2,  is  a  compound  ratio. 


288  RATIO    AND    PROPOIiTION. 

The  simple    ratio  of  12  :  4,    or  J/,       is    3; 

"      "      8  :  3,     "    f ,        "     4  • 

"  compound  "      ''    ^8  :'  2  [  '''  "^  ^  t  "  1^'  ^^  ^  times  4, 


the  product  of  the  two  simple  ratios  which  form  the  compound 
ratio. 

525*    Principles  of  llatio. 

I.  Tlxe  terms  of  a  ratio  must  he  of  the  same  kind  or  denom- 
illation. 

II.  In  an  expressed  ratio,  the  antecedent  is  a  dividend  and 
the  consequent  is  a  divisor. 

in.  TJie  ratio  is  the  quotient  of  the  antecedent  divided  by  the 
consequent. 

lY.  Uie  product  of  any  two  or  more  ratios  equals  the  ratio 
of  their  products  ;  and 

V.  A  compound  ratio  is  reduced  to  a  simple  one,  by  multi- 
plying all  the  antecedents  together  for  a  new  antecendent,  and 
all  the  consequents  together  for  a  new  consequent. 

EXETtCISES. 

1.  Express  the  ratio  18  to  3,  and  of  3  to  18;  of  20  to  4,  and  of 
4  to  20. 

2.  Express  in  both  forms  the  ratio  of  35  to  5  ;  of  28  to  2  ;  of  43 
to  7  ;  and  of  9  to  15. 

3.  Find  the  ratio  of  18  to  6  ;  of  21  to  3  ;  and  of  14  to  3. 

4.  The  antecedent  is  16f ,  and  the  consequent  is  3^.  What  is  the 
ratio  ? 

5.  Wliat  is  the  ratio  in  eacli  of  the  expressions  5G  :  8 ;  $51  :  $3; 
l^  yd. :  ^  yd.  ? 

6.  Find  the  ratio  of  16|  to  13f ;  of  3.5  :  .7 ;  of  .25  :  3.75  ;  and  of 
5.7  :  8.4. 

7.  Express  the  compound  ratio  of  j  ^  .'  ^  ^  >  . 

8.  The  antecedents  of  a  compound  ratio  are  5,  73,  and  30,  and 
the  consequents  are  8,  3,  and  5.     Express  the  compound  ratio. 

10,800  :  120  =  DO. 


SIMPLE    PROPOIITION.  289 

SECTION   II. 

526.  ^roporHo7l  is  an  equality  of  ratios. 

527.  Simple  Proportion  is  an  equality  of  two  simple 
ratios. 

528.  The  Sig7i  of  Proportion  is  the  double  colon  (: :).  It 
is  written  between  the  ratios,  and  is  read  "  as,"  or  "  equals.'' 

Example. — The  ratios  12  :  6  and  8  :  4  being  equal,  form 
the  proportion  12  :  G  :  :  8  :  4,  which  is  read,  "  12  is  to  6  as 
8  is  to  4,"  or  "  the  ratio  of  12  to  6  equals  the  ratio  of  8  to  4." 

Proportion  may  also  be  expressed  by  the  sign  of  equality. 
Thus,  12  :  6  :  :  8  :  4  may  be  written  12  :  6  ==  8  :  4,  or  ^§-  =  f . 

529.  The  Extremes  of  a  proportion  are  the  first  and 
fourth  terms  ;  and 

530.  The  Means  are  the  second  and  third  terms. 

531.  We  may  write  the  proportion  12  :  6  :  :  8  :  4  in  the 
fractional  form,  and  reduce  the  fractional  expressions  to 
similar  fractions  ;  ^i^ :  :  |  =  4|  =  =  li  ^^  W  *  =  1^4*  ^^^ 
factors  12  and  4,  of  the  first  numerator,  are  the  extremes 
of  the  proportion  ;  and  the  factors  6  and  8,  of  the  second 
i^merator,  are  the  means  ;  and  the  products  of  these  two 
sets  of  factors  are  equal.    Hence,  *" 

532.    Principles  of  Proportion, 

I.  The  product  of  the  extremes  equals  the  product  of  the 
means. 

II.  The  first  term  is  greater  or  less  than  the  second,  according 
as  the  third  term  is  greater  or  less  than  the  fourth. 

533.  Ex.  1.  What  is  the  fourth  term  of 
the  proportion  21  :  6  :  :  14  :  —  ?  solutiok. 

Explanation.  —  84,    the    product  of    the     <?|f ^  =  |-f  =i 
means,  6  and  14,  is  also  the  product  of  the 
extremes  (532, 1.).     We  now  have  84,  the  product  of  two 
13 


290 


RATIO    AND    PROPORTION. 


factors,  and  21,  one  of  the  factors,  to  find  the  other  factor, 
which  we  do  by  dividing  84  by  21.  The  result,  4,  is  the 
fourth  term  required. 

Ex.  2.  Find  the  second  term  of  the  pro-  solution. 

portion  15  :  —  :  :  30  :  12.  '^■r^-='^=0 

Explanation. — Since  the  product  of  the  extremes  equals 
the  product  of  the  means,  we  multiply  the  extremes,  15 
and  12,  together,  and  divide  the  product,  180,  by  the 
known  mean,  30.   The  result,  6,  is  the  term  required.  Hence, 

I.  The  product  of  the  means  divided  by  either  extreme  gives 
the  other  extreme  ;  and 

II.  ITie  product  of  the  extremes  divided  by  either  mean  given 
the  other  mean. 

PJt  OBZE3IS. 

Find  the  unknown  term  in  each  of  the  following  proportions . 


—  :5. 


1.  10    :  8    : 

:  15  : 

3.     7^:61: 

:  —  : 

5.  13    :  18  : 

:18: 

7.    4    :    7: 

:  —  : 

9.  .35    :  — : 

:  .3: 

6. 
2.4. 


2.  27 

4.  — 

6.  6 

8.  — 

10.  15 


:  —  ; 

13|: 

:21  : 

15    : 

:  —  : 

12    : 

;    9: 

■I-      : 
10    : 

5. 
35. 


534.  Ex.  If  16  tons  of  hay  cost  $212,  how  much  will  11 
tons  cost? 


Explanation. — 1st. 
Finding  the  ratios. — 
The  ratio  of  16  T.  to 
11  T.  must  be  the 
same  as  that  of  $212, 
the  cost  of  16  T.,  to 
$— ,  the  cost  of  11  T. 
The  16  T.  and  11  T. 
must  therefore  form 


T/ie  raiioa. 


Statements. 


Finding 
unknown 
term. 


60LTTTI0N. 

11  T. 

16  T. :  11  T.  i 
11  T. :  16  T. : 


C 16  T. 

I  $212 

) 


$53 


11  X 


$583 


=  $U-H 


one  couplet  of  the  proportion,  and  the  $212  and  the  $ — 
the  other  couplet. 

2d.  Making  the  statement. — Since  16  T.  are  more  than  11  T., 
$212,  the  cost  of  16  T.,  must  be  more  than  $ — ,  the  cost  of 


SIMPLE    PROPORTION.  291 

11  T.  Therefore,  if  we  write  16  T.  :  11  T.  for  the  first  ratio, 
we  must  write  $212  :  $ —  for  the  second  (see  532, 11. )  ;  or,  if 
we  write  11  T.  :  16  T.  for  the  first  ratio,  we  must  write  $ —  : 
$212  for  the  second. 

od.  We  find  the  unknown  term  by  533, 1,  or  IT. 

For  convenience  in  making  the  statement,  we  take  the 
two  given  numbers  of  the  same  kind,  for  the  terms  of  the 
first  ratio,  and  the  other  given  number  for  the  first  term  of 
the  second  ratio,  or  the  third  term  of  the  proportion.  Now, 
writing  this  third  term,  and  applying  (532,  II.),  we  readily 
determine  which  term  of  the  first  couplet  to  write  for  the 
antecedent,  and  which  for  the  consequent.     Hence, 

535»    Hule  for  Computations  hi  Simple  !p7'oportio7i. 

I.  For  the  third  term,  write  the  number  that  is  of  the  same 
kind  as  the  unknown  term. 

n.  Write  the  other  two  numbers  as  the  first  ratio,  putting  the 
greater  for  the  first  term,  when  the  unknown  term  is  to  be  less 
than  the  third  term  ;  and  for  the  second,  when  it  is  to  be  greater. 

m.  For  the  fourth  or  unknown  term,  multiply  the  second 
and  third  terms  together,  and  divide  the  product  by  the  first  term. 

Notes. — 1.  Wheji  the  terms  of  the  first  ratio  are  of  different  denomina- 
tions, they  must  be  reduced  to  the  same  denomination  before  multiplying 
and  dividing. 

2.  Cancel  all  like  factors  from  the  given  extreme  and  either  of  the  means. 

See  Manual. 

PMOBZiJEMS. 

II.  If  53  bu.  of  lime  cost  $13.25,  how  nmch  will  29  bu.  cost  ? 

12.  If  my  store  rent  is  $315  for  7  mo.,  how  much  is  it  for  2  yr. 
5  mo.?  $1,305. 

13.  If  12|  lb.  of  beef  cost  $1.78|,  how  much  will  5-*-  lb.  cost.  ? 

14.  How  much  is  a  cental  of  wheat  worth,  at  $1.40  per  bushel  ? 

15.  How  much  will  9  rm.  2  quires  of  note  paper  cost,  if  3  rm.  14 
quires  cost  $4.62^?  $11.37^, 


292  RATIO    AND    PROPORTION. 

16.  The  driving-wheel  of  a  locomotive  makes  401  revolutions  in 
running  1  mi.  20  rd.  4  ft.  How  many  revolutions  will  it  make  in 
running  23  mi.  16  rd.  12  ft.  ?  8,694. 

17.  If  31  trees  bear  271  bu.  1  pk.  of  apples,  how  many  apples 
will  56  trees  bear,  at  the  same  rate  ?  4^0  hi. 

18.  After  cleansing  475  pounds  of  wool  for  manufacture,  it 
w^eighed  only  266  pounds.  At  the  same  rate,  how  much  would 
645  pounds  weigh,  after  cleansing  ? 

19.  If  a  Mississippi  River  steam-boat  runs  779  mi.  in  3  da.  10  h., 
how  far  will  she  run  in  4  da.  3  h.  ?  940 ±  mi. 

20.  If  18f  cd.  of  wood  cost  $81.75,  how  much  will  21.25  cd.  cost  ? 

21.  When  8.7  tons  of  hay  cost  $73.95,  how  much  must  be  paid  for 
9.75  tons  ?  $82.87±. 

22.  If  a  cheese  weighing  53  lb.  3  oz.  costs  |6.28,  how  much  will 
a  cheese  cost  that  weighs  44|-  lb.  ? 

23.  The  interest  on  a  certain  sum  of  money  for  1  yr.  5  mo.  22  da. 
is  $46.55.  What  is  the  interest  on  the  same  sum  for  11  mo. 
12  da.  ?  $29.92^. 

24.  If  a  pole  6  ft.  high  casts  a  shadow  7|-  ft.  long,  how  high  is  a 
tree  whose  shadow  is  85  ft.  long  ?  68  ft. 


SECTION  III. 

coMTOzrjvs)   'P^o^o^Tiojsr. 

586i  Compou7id  Proportion  is  an  equality  of  a  com- 
pound and  a  simple  ratio,  or  of  two  compound  ratios ;  as, 

^''^\..^(\.(K.^^    5:9)        (5:2 
-^2^^|-::10:6,andj^2:4[-i2:3 

537.  Since  in  a  simple  proportion  the  product  of  the  ex- 
tremes equals  the  product  of  the  means  (see  532,  I.)  ;  and 
since  a  compound  ratio  is  reduced  to  a  simple  one,  by  mul- 
tiplying all  the  antecedents  together  for  a  new  antecedent, 
and  all  the  consequents  together  for  a  new  consequent 
(see  ^%^i  v.),  it  foUows  that 


COMPOUND    PROPORTION.  293 

In  a  compound  proportion  the  product  of  the  extremes  equals 
the  product  of  the  means. 

538i  Ex.  If  4  men  in  8  days  cut  48  cords  of  wood,  how 
many  cords  will  5  men  cut  in  6  days  ? 

Explanation.  —  Since  bolution. 

the  unknown    term   is      ^  ^^^  •  ^  ^^^^  I ; ;  ^  cd. :  -  cd. 

J  -i.    i-i,     ^o      ^  <^«2/s  ••  6  days  f  •  •  -^  '^""  • 

cords,  we  write  the  48  ^    •' 

cords  for  the  first  term  3 

of  the  second  ratio,  or  ^     ^^ 

the  third  term  of  the  ^^^^^^^'=4Jj  cd. 

proportion,   as  in  535.  ^^ 

Since  the  unknown  term 

depends  upon  both  the  number  of  men  and  the  number  of 

days,  we  arrange  the  other  given  numbers — 4  men  and  5 

men,  8  days  and  6  days — as  the  terms  of  a  compound  ratio, 

observing  the  same  directions  in  writing  the  terms  of  each 

couplet,  as  in  writing  the  first  ratio  of  a  simple  proportion. 

Thus,  4  men  cut  48  cords,  5  men  will  cut  more  ;  hence, 

4  men  :  5  men  ;  and  in  8  days  they  cut  48  cords,  in  6  days 

they  wiU  cut  less ;  hence,  8  days  :  6  days.     Having  made 

the   statement,  we   multiply  the  second  and  third  terms 

together,  and  divide  the  product  by  the  first  term,  as  in 

simple  proportion.     The  quotient,  45  cords,  is  the  fourth  or 

unknown  term.     Hence, 

539.  ^ule  for  Compteiatlo7is  in  Co7npou7id  Proportion, 

I.  For  the  third  term,  write  the  number  that  is  of  the  same 
kind  as  the  unknown  terra. 

'  n.  Write  each  two  of  the  other  numbers  that  are  of  the  same 
hind,  as  a  couplet  of  the  compound  ratio,  observing  the  same 
directions  as  in  writing  the  tei'ms  of  the  first  ratio  in  a  simple 
proportion. 

HI.  For  the  fourth  or  unknown  term,  multiply  the  second  and 
third  terms  together,  and  divide  the  product  by  the  first  term. 

See  Manual. 


294  RATIO    AND    PROPORTION. 

Find  the  unknown  term  in  each  of  the  following  four  proportions : 
1.4.    8f--^-      •     '^^  .25yd.wide:^yd.widef  ••^^^•^^^• 


25:      5 


:io^[::-:7f  4. 


—  :14 


4.5  :  .6 

5.  If  the  carpet  for  a  room  15  ft.  x  16  ft.  costs  $40,  how  much 
will  a  carpet  of  the  same  kind  cost  for  a  room  14  ft.  x  18  ft.  ?     $^2. 

6.  A  flouiing  mill  running  10  hours  per  day,  makes  1,365  barrels 
of  flour  in  13  days.  How  many  barrels  will  the  same  mill  make  in 
39  days,  running  16  hours  per  day  ? 

7.  I  sold  a  village  lot  5|-  rods  front  by  7|-  rods  deep,  for  |580 ; 
and  another  lot  4f  rods  front  by  ^\  rods  deej),  at  the  same  rate. 
How  much  did  I  receive  for  the  second  lot  ?  $622.22. 

8.  A  pile  of  4-foot  wood  2,450  ft.  long  and  6  ft.  high,  was  drawn 
to  a  R.R.  station  by  15  teams  in  14  days.  At  the  same  rate,  how 
many  days  will  it  take  24  teams  to  draw  a  pile  2,016  ft.  long  and  5 
ft.  high  ?  6. 

9.  If  5,280  rails  will  build  360  rods  of  8-rail  fence,  how  many 
rails  will  be  required  to  build  276  rods  of  10-rail  fence  ? 


SECTION  IV. 


540.  A  ^arlners?ii2y y  or  a  Company y  is  an  associa- 
tion of  two  or  more  persons  for  the  transaction  of  business 
as  an  individual. 

541.  A  ^irm  is  the  name  under  which  a  company  trans- 
acts business. 

Notes. — 1.  Each  member  of  a  eompany  or  firm  is  a  Partner. 

2.  A  firm  is  sometimes  called  a  House  ;  as  the  House  of  Jaj'  Cooke  &  Co. 

542.  ^rofiis  are  the  gains  to  be  shared  among  the 
members  of  a  firm  ;  and 

543.  Assessments  are  sums  to  be  paid  by  members  of 
a  firm,  to  meet  expenses  or  cover  losses. 


PARTNERSHIP,  295 

544.  Capital  is  the  time,  or  the  money,  or  both,  in- 
vested in  business. 

545.  A  Simple  (Partnership  is  one  in  which  each 
partner's  share  in  one  of  the  elements  of  capital — time  or 
money — is  the  same  ;  and 

546.  A  Compound  ^art7iership  is  one  in  which  the 
l)artners'  shares  in  one  or  both  elements  of  capital  vary. 

CJLSE     I. 
Simple  Partnership. 

547.  If  one  partner  furnishes  3  times  as  much  money  as 
another,  or  if  he  furnishes  the  same  amount  of  money,  and 
it  remains  in  business  3  times  as  long,  his  share  of  the 
profits  ,and  losses  is  3  times  as  great  as  that  of  the  other 
partner.     Hence, 

I.  When  the  time  is  equal,  the  profits  and  losses  are  shared  by 
the  partners  in  proportion  to  their  respective  shares  of  the  money 
in  business. 

II.  When  the  shares  of  money  are  equal,  the  profits  and 
losses  are  shared  by  the  partners  in  proportion  to  the  respective 
times  their  money  is  in  business. 

548.  Ex.  1.  A,  B,  and  C  enter  into  partnership,  A  fur- 
nishing $3,500,  B  $2,500,  and  C  $2,000  of  the  capital.  Their 
profits  are  $3,200.     What  is  each  man's  share  ? 

Explanation.  — 
Since  the  partners         ^^  _^      ...    '°''''"''!; 
furnish  Lerent        ^^^^^^^  +  ^'^^'^  +  ^'-^'^  =  ^^^^^^^ 


S3, 500  : :  $3,200  :  As  share. 
$2,500  ::  $3,200  iB's     " 
$2,000  ::  $3,200  :  G's     " 


amounts  of  money  $8,000 

for   the   same  $8,000 

length  of  time,  $8,000 

they  wiU  share  in 

the  profits  in  pro-  Hence,  A's  share  is  $l,JfOO  ;  B's  share^ 

portion    to    their  $1,000 ;  and  C's  share,  $800. 

respective    shares 

of  the  money  in  business  (547,  I.).     Adding  their  shares  of 


296  RATIO    AND    PROPORTION. 

money  furnished,  we  have  $8,000,  the  entire  capital.  Then, 
Whole  capital :  each  mail's  capital ::  whole  gain :  each  marl's  share  of  gain. 
Finding  the  unknown  term  in  each  of  the  three  proportions, 
we  have  each  man's  share  of  the  gain. 

Ex.  2.  D  and  E  formed  a  partnership  for  1  year,  each 
furnishing  the  same  capital.  At  the  end  of  8  months,  D 
drew  out  his  capital,  his  interest  continuing  to  the  close  of 
the  year,  when  the  profits  were  $4,500.  What  was  each 
one's  share  ? 

Explanation. —  colution. 

Since   the  partners  8  mo.  +  12  mo.  =  20  mo. 

furnish    the   same 

amount   of    money,      20  mo.  :    8  mo.  : :  $Jf,,500  :  D's  share, 
for  different  lengths      20  mo.  :  12  mo.  : :  $4,500  :  E^     " 
of    time,    they    will 

share  in  the  profits  Hence,  D's  share,  $1,800  ; 

in  proportion  to  the  E's  share,  $2, 700. 

respective      times 

their  money  is  in  business  (547, 11.).    Adding  their  shares  of 

time,  we  have  8  mo.  (D's  share  of  time)  + 12  mo.  (E's  share 

of  time)  =  20  mo.,  the  sum  of  their  shares  of  time.     Then, 

Whole  time :  each  man's  time : :  whole  gain :  each  mart's  share  of  gain. 

PM  OBILBMS. 

1.  A  and  B  form  a  partnershiiD.  A  furnishes  $2,500  of  the 
capital,  and  B  $4,000,  and  they  gain  $1,950.  What  is  each  man's 
share  ?  A's,  $750;  B's,  $1,200. 

2.  Three  men,  having  1  cow  each,  hire  a  pasture  for  the  season, 
for  $22.50.  A's  cow  is  in  the  pasture  4  months,  B's  cow  6  months, 
and  C's  cow  5  months.     How  much  does  each  man  pay  ? 

3.  A  factory,  insured  in  the  ^tna  Ins.  Co.  for  $5,000,  in  the  Home 
Ins.  Co.  for  $7,500,  in  the  Continental  Ins.  Co.  for  $6,000,  and  in  the 
ISToilh  American  Ins.  Co.  for  $4,500,  was  damaged  by  fire  to  the 
amount  of  $10,120.     How  much  of  the  loss  fell  upon  each  Co.  ? 

^tna,  $2,200;  Borne,  $3,300; 
Gantinentalf  $2,640;  JV.  A.,  $1,980. 


PARTNERSHIP. 


297 


4.  A,  B,  and  C  bought  some  city  lots  in  company,  A  fumisliing 
$1,800  of  the  purchase  money,  B  $1,500,  and  C  $1,200.  They  sold 
the  lots  for  $8^420.     How  much  did  each  man  gain  ? 

C^SE     II. 
Compound  Partnership. 

549.  If  one  partner  furnishes  3  times  as  much  capital  as 
another,  either  in  money  or  in  the  time  his  money  remains 
in  the  business,  or  in  the  product  of  money  and  time,  his 
share  of  the  profits  and  losses  is  3  times  as  great  as  that  of 
the  other  partner.    Hence, 

The  product  of  each  man's  capital  multiplied  by  the  time  it  is 
in  business,  represents  his  proportionate  share  of  the  capital. 

550.  Ex.  A,  B,  and  C  form  a  partnership  for  8  months, 
A  putting  in  $2,000,  B  $3,000,  and  C  $1,500.  If  C  draws 
out  his  capital  in  4  months,  and  B  his  in  6  months,  and  the 
profits  are  $5,600,  what  is  each  man's  share? 

Explanation. — 
Since  the    times  solution. 

and  shares  of 
money  are  both 
unequal,  each 
man's  share  of  the 
profits  depends 
both  upon  the 
amount  of  money 
furnished  by  him 
and  the  time  it  is 
used  in  the  busi- 
ness. (See  549.) 
A's  money,  $2,000, 

for  8  months,  is  the  same  as  8  times  $2,000,  or  $16,000,  for 
1  month  ;  B's  money,  $3,000,  for  6  months,  is  the  same  as 
$18,000  for  1  month  ;  and  C's  money,  $1,500,  for  4  months, 
is  the  same  as  $6,000  for  1  month.     The  whole  proportion- 


$2,000  for  8  mo.  =  S  16,000  for  1  mo. 


3 MO 
1,500 


$1^0,000 
$lfi,000 
$}fi,000 


$16,000 

$18,000 

$6,000 


=     18,000   " 
=       6,000   " 

$4.0,000 


$5,600  :  A' s  share, 
$5,600  :  B's     " 
$5,600  :  G's     « 


Hence,  A's  share  is  $2,2^0  ; 
B's,  $2,520;  0%  $8^0, 


298  RATIO    AND    PROPORTION. 

ate  capital  is  $40,000,  of  whicli  A's  share  is  $16,000 ;  B's, 

$18,000  ;  and  C's,  $6,000.     Then, 

Proportionate  Capital :  ea^h  man's  sha/re : :  whole  gain :  each  man's  gain. 

551.  Upon  the  principles  deduced  in  547,  549,  are  based 

the 

^utes  for   Computations    i7i    (Partners ?iip, 

I.  For  Simple  Partnership. 

1.  For  the  whole  capital,  add  all  the  partners'  shares  of 
money  or  time. 

2.  For  each  partner's  share  of  the  gain  or  loss,  state  a  pro- 
portion, thus : —  Whole  capital :  each  partner's  capital  : :  whole 
gain  or  loss  :  each  partner's  gain  or  loss. 

IL  For  Compound  Partnership. 

1.  For  each  partner's  proportionate  share  of  the  capital, 
multiply  his  money  by  the  time  it  is  in  business. 

2.  For  the  whole  capital,  add  all  the  proportionate  shares. 

3.  For  each  partner's  share  of  the  gain  or  loss,  state  a  pro- 
portion, thus : —  Whole  proportionate  capital :  each  partner's  pro- 
portionate share : :  whole  gain  or  loss :  each  partner's  gain  or  loss. 

FROBIjEMS. 

5.  A  and  B  engage  in  trade  together,  A  putting  in  $1,500  for  9 
months,  and  B  $2,500  for  6  months.  They  gain  $2,394.  What  is 
each  one's  share  ?  A's^  $1,134;  -S'«,  $1,260. 

G.  The  firm  of  Sanford,  Wright,  &  Thomas  manufacture  agricul 
tural  implements,  and  their  capital  is  $12,000,  of  which  Mr.  S.  fur- 
nished $5,500,  Mr.  W.  $4,500,  and  Mr.  T.  $2,000.  Last  year  they 
made  $9,360.    What  were  each  partner's  profits  ? 

7.  Three  men  harvested  and  thrashed  a  field  of  grain  on  shares, 
A  furnishing  4  hands  5  days,  B  6  hands  4  days,  and  C  5  hands 
8  days.  The  whole  crop  was  630  bushels,  of  which  they  had  ^. 
How  much  did  each  receive  ?  A,  30  lu.;  B,  36  hi.;  C,  60  hi. 

8.  A,  B,  and  C  are  the  partners  in  a  store.  A  furnishes  $2,300 
for  1  yr.,  B  $1,750  for  10  mo.,  and  C  $1,450  for  1  yr.  3  mo.,  and  they 
lose  $472.50.    What  is  each  man's  loss  ? 


REVIEW     PROBLEMS.  299 

SECTION  V. 

1.  The  property  of  an  insolvent  debtor  amounts  to  $6,343,  and 
his  liabilities  to  $17,550.  How  much  will  a  creditor  receive  on  a 
debt  of  $1,250? 

2.  If  .1875  bu.  of  sweet-potatoes  cost  $.30,  what  will  be  the  cost 
of  .875  bu.  ?  $l.JtO. 

3.  A  man  whose  estate  is  worth  $19,250,  directs,  by  his  will,  that 
his  property  shall  be  so  divided  among  his  four  children  that  his 
daughter  shall  receive  $4,  his  youngest  son  $5,  and  his  second  son 
$6,  as  often  as  his  eldest  son  receives  $7.  How  much  will  each 
child  receive?  Daughter,  $3,500 ;  youngest  son,  $4^375 ; 

second  son,  $5,250;  eldest  son,  $6,125. 

4.  If  400  ft.  of  flooring  1|^  in.  thick  are  required  for  a  room, 
how  much  flooring  1|^  in.  thick  will  be  required  for  a  room  3  times 
as  long  and  2  times  as  wide  ?  2,880  ft. 

5.  Four  men  paid  $13  for  a  carriage  to  convey  them  from  a  R.R. 
station  to  their  homes,  which  were  distant  16  mi.,  24  mi.,  28  mi, 
and  36  mi.  respectively.  They  paid  in  proportion  to  the  distances 
they  rode.     How  much  did  each  man  pay  ? 

0.  The  Interest  of  $286.25  for  a  certain  time  is  $27.48.  What  is 
the  interest  of  $59.50  for  the  same  time  ? 

7.  A  bin  12  ft.  x  7  ft.  x  6  ft.  will  hold  405  bushels  of  grain. 
How  many  bushels  will  a  bin  hold  that  is  8  ft.  x  7  ft.  x  4  ft.  ?     180. 

8.  If  17f  cords  of  wood  will  produce  745|-  bushels  of  charcoal, 
how  many  cords  of  wood  will  be  required  to  produce  l,677f 
bushels  ? 

9.  A  and  B  bought  a  mill,  A  paying  $2,400,  and  B  $3,200.  8 
months  afterward  C  bought  f  of  A's  share,  and  D  bought  f  of  B's 
share.  The  profits  of  the  mill  for  the  year  were  $5,040.  How  much 
was  each  man's  share  ?  A,  $1,872;  B,  $2,280;  G,  $288;  2>,  $600. 

10.  Three  men  shipped  a  cargo  of  1,500  barrels  of  flour  to 
England,  A  furnishing  700  barrels,  B  200  barrels,  and  C  the 
balance.  In  a  storm  195  barrels  were  thrown  overboard.  How 
should  the  loss  be  shared  among  the  owners  ? 


SECTION  I. 

^BFIJ^ITIOJVS    ;dJV:D    JVOTdTIOJST, 
552$  A  !^ool  of  a  number  is  one  of  its  equal  factors. 
553i  The  Square  ^oot  of  a  number  is  one  of  its  two 
equal  factors. 

,    554.  The   Ctcde  !Eoot  of  a  number  is  one  of  its  tliree 
equal  factors. 

Example. — 3  is  a  root  of  9,  of  27,  and  of  81,  because  9  =  3 
X  3  or  32,  27  =:  3  X  3  X  3  or  33,  and  81  =  3  X  3  X  3  X  3  or 
3^     3  is  the  square  root  of  9,  and  the  cube  root  of  27. 

Notes.— 1.  One  of  the  four  equal  factors  of  a  number  is  its  Fourth  Boot, 
one  of  the  five  equal  factors  is  its  Fifth  Boot,  and  so  on. 

2.  A  number  whose  square  root  can  be  obtained,  is  a  Perfect  Square; 
and  one  whose  cube  root  can  be  obtained,  is  a  Perfect  Gvhe. 

555*  l7ivo2uHon  is  the  process  of  finding  any  required 
power  of  a  number.     (See  82-87). 

556.  SJvottcHoji^   or  ^xiracHo7i  of  ^oois,  is  the 

process  of  finding  any  required  root  of  a  number. 

Note.— Involution  and  evolution  are  converse  operations. 

557.  JEJxtracHon  of  Square  ^ooti^  the  process  of 
finding  one  of  the  two  equal  factors  of  a  number;  and 

558.  J^xiracHo7i  of  Citbe  !Eoot  is  the  process  of 
finding  one  of  the  three  equal  factors  of  a  number. 

559.  The  Sig7Z  of  Square  ^oot  is  V,  called  the 
Radical  Sig7i;  and 

560.  The  Sig7i  of  Czibe  !Soot  is  V.  Thus,  V/64  is 
read  "  The  square  root  of  64 ; "  and  1/125  is  read  "  The 

cube  root  of  125."      See  Manual. 
Note.- A  Surd  is  an  indicated  root  which  can  not  be  obtained ;  as  ^5,  y  7. 


EXTllACTION    OF    SQUARE    ROOT.  301 

SECTION  II. 

561.  The  least  and  the  greatest  number  that  can  be  ex- 
pressed by  one  figure  are  1  and  9  ;  by  two  figures,  10  and 
99  ;  by  three  figures,  100  and  999  ;  and  so  on. 

The  squares  of  these  numbers  are 

1^=  1        10^=    100        100^=  10,000 

9^=81        99^=9,801        999'=998,001     and  so  on. 

By  examining  these  numbers  and  their  squares,  we  see 
that 
The  square  of  a  number  expressed  by  consists  of 

One  figure  one    or    two  figures ; 

Two  figures  three  "     four      " 

Three    "  five     "     six        " 

Four     "  seven "     eight     " 
and  so  on.    That  is, 

I.  The  square  of  any  number  consists  of  twice  as  many 
figures  as  the  number,  or  one  less, 

II.  If  a  number  be  separated  into  periods  of  two  figures  each, 
beginning  with  ones,  its  square  root  will  consist  of  as  many 
figures  as  there  are  full  and  partial  periods  in  the  number. 

562,  If  we  write  any  digits,  as  2  and  9,  successively  as  ones, 
tens,  hundreds,  and  so  on,  and  square  them,  we  shall  have 

2'=  4  20'=    400  200'=  40,000 

9'=81  90'=8,100  900^=810,000 

By  examining  these  numbers,  we  see  that 

The  square  of  the  ones         is  in  the  first  period ; 
"        "  "       tens  "       second  " 

"        «  "       hundreds       "       third     « 

and  so  on.     That  is. 
The  square  of  the  left-hand  figure  of  a  root  is  wholly  in  the 
left-hand  period  of  the  number  or  power. 


302  EVOLUTION. 

563t  If  we  square  any  numbers  expressed  by  two  figures, 
as  20  and  25,  60  and  63,  90  and  99,  we  shall  have 

20^=400  00^=3,600  90^^=8,100 

35^=625  63^=3,969  99'^=9,801 

By  comparing  these  roots  and  their  squares,  we  see  that 
4  is  the  greatest  square  in  6,  the  hundreds  of    625 ; 
36     "  "  "         39,    "  "  3,969; 

81     "  "  "        98,    "  "  9,801.    That  is, 

The  greatest  square  in  the  left-hand  period  of  a  number  is  the 
square  of  the  left-hand  figure  of  the  root. 

564.  "We  will  now  square  the  number  37,  for  the  purpose 
of  learning  of  what  parts  the  square  is  composed. 

Ex.   37=30  +  7,  and  3^=30  +  7  multipHed  by  30  +  7. 

Explanation.  — The  square  ^^^^^^^^ 

of  the  ones =49  ;  the  product  3  0-h    7  =        8  7 

of  the  tens  by  the  ones  (7x30)  3  0+    7  =        3  7 

+  the  product  of  the  ones  by  2 10+A9  =     25  9 

the  tens  (30  x  7),  or  two  times      900+210  111 

the  product  of  the  tens  and      9 00+JL20+Jt.9  ~  1369 
ones =420  ;  the  square  of  the 

tens  =  900  ;  and  the  sum  of  these  three  partial  products = 
1,369.    Hence, 

The  square  of  a  number  consisting  of  two  figures,  is  equal  to 
the  square  of  the  tens,  plus  two  times  the  product  of  the  tens 
and  the  ones,  plus  the  square  of  the  ones. 


565.  Ex.1.  What  is  the 


FIRST  SOLUTION. 


square  root  of  1,369  ?                        13-6  9 
Explanation.  —  Separ-  _l 


ating    the    number    into     Di'^idend.    j^6 

periods    of    two    figures  -^^ 

each,    we    find  that  the 

square  root  will  consist  of  two  figures.     (561,  II.) 


3  7  Root 

6  0  Trial  divisor. 

7_ 

6  7  Compute  divisor. 


EXTRACTION    OF    SQUARE    ROOT. 


303 


6EC0ND  SOLUTION. 


13-6  9 

9 


U69 

Jf69 


37 


67 


Since  9  is  the  greatest  square  in  13,  the 
first  period,  we  write  3,  its  square  root,  for 
the  first  figure  of  the  root  (563).  Taking 
9,  the  greatest  square,  from  the  left-hand 
period,  and  annexing  69,  the  next  period, 
to  the  remainder,  we  have  469.  This  num- 
ber is  made  up  of  two  times  the  product  of  the  tens 
and  ones  of  the  root,  plus  the  square  of  the  ones  (564)  ; 
i,  e.f  of  30  (=3  tens)  x  2  x  the  ones  of  the  root,  +the  square 
of  the  ones.  Dividing  469  by  the  trial  divisor,  60  (=2 
times  3  tens,  or  30),  we  obtain  7,  which  we  write  for  the 
second  figure,  or  ones,  of  the  root.  Since  60—2  times  3 
tens,  and  469=2  times  3  tens  x  the  ones  +  the  square  of  the 
ones,  we  add  7  to  the  trial  divisor,  60,  making  67.  Then 
multiplying  67,  the  complete  divisor,  by  7,  the  last  figure  of 
the  root,  we  obtain,  1st,  7  times  7  =  the  square  of  the  ones ; 
and  2dj  7  times  60  =  2  times  3  tens  x  7  ones  =  2  times  the 
product  of  the  tens  and  the  ones.  The  product,  469,  is  the 
same  as  the  dividend,  and  37  is  the  square  root  required. 
.  In  the  Second  Solution  we  have  placed  the  quotient 
figure,  7,  in  the  place  of  the  0  in  the  trial  divisor,  thus  com- 
pleting the  divisor  at  once. 

BAT.TTTTDW. 

Ex.  2.  What  num- 
ber is  the  square  root 
of  555,025? 

Explanation.  —  In 
extracting  the  square 
root  of  a  number,  only 
two  periods  of  figures 
are  considered  at  once.  Therefore,  in  obtaining  any  figure  of 
the  root,  after  the  first,  we  regard  the  figure  or  figures  of 
the  root  already  found  as  tens,  and  the  figure  sought  as 
ones,  and  find  each  succeeding  figure  in  the  same  manner 
as  we  find  the  second  figure  of  a  root  consisting  of  two 
figures,  as  will  be  seen  in  the  Solution. 


1st  dividend. 


2d  dividend. 


55-5  0-25 
49 

650 

576 

7U25 
71,25 

7  If  5  Root. 


1  If.  If  1st  divisor. 


Ilf85  2(Z  divisor. 


304  EVOLUTION. 

SOLUTION. 

2  7,85 


Ex.   3.     Find    the    square   root   of  ^olu 

748.0225.  7^Jf8.0  2'^. 

Explanation. — Separating  the  num- 

ber  into  integral  and  decimal  j)eriods,    "f  |  ^ 
by  counting  left  and  right  from  ones, 


we  proceed  as  in  Ex.  1  and  2,  putting       ip^o 
a  decimal  point  before  the  figure  of  the 


5^8 


5^65 


root  obtained  from  using  the  first  deci-         2782  5 
,        .    ,                        ^                                  27825 
mal  period.  — 

Ex.  4.  Extract  the  square  root  of  g^^y. 

Explanation. — Since  a  frac-  solution. 

tion  is  squared  by  squaring      ^^  =  ^  ^  ^57^  ==  ^^ 
each    term    separately   (335), 

and  since  evolution  is  the  converse  of  involution  (556,  Note), 
we  extract  the  square  root  of  each  term  separately. 

566.  Upon  the  principles  deduced  in  561-564,  is  based 
the 

!%tile  for  J^xtraciion  of  Square  ^oot, 

I.  To  determine  the  number  of  figures  in  the  root. 
Separate  the  numher  into  periods  of  tivo  figures  each,  count- 
ing  left  and  right  from  ones. 

n.  For  the  first  figure  of  the  root. 

1.  Find  the  root  of  the  greatest  square  in  the  left-hand 
period,  for  the  first  figure  of  the  root. 

2.  Subtract  this  square  from  the  first  period;  and  to  the 
remainder  annex  the  next  period,  for  the  first  dividend. 

m.  For  the  second  figure  of  the  root. 

1.  Double  the  root  already  found,  considered  as  tens,  for 

the  first  trial  divisor,  by  which  divide  the  first  dividend;  and 

write  the  result  for  the  second  figure  of  the  root,  and  also  in  the 

place  of  ones  in  the  trial  divisor,  thus  forming  the  complete 


2.  Multiply  the  complete  divisor  by  the  second  figure  of  the 


EXTRACTION    OF    SQUARE    ROOT.  305 

root;  subtract  the  product  from  the  Jirst  dividend;  and  to  the 
remainder  annex  the  next  period  for  a  new  dividend. 

rV.  For  tlie  succeeding  figures  of  the  root. 
Proceed  with  the  second,  and  with  each  succeeding  dividend, 
in  the  same  manner  as  with  the  first,  until  all  the  periods  are  used. 

Notes.— 1.  If  any  dividend  is  less  than  tlie  divisor,  annex  a  cipher  to  the 
root,  and  also  to  the  divisor,  and  annex  the  next  period  to  the  dividend,  for 
a  new  dividend. 

2.  If  there  is  a  remainder  after  all  the  periods  have  been  used,  i.  e.,  in  ex- 
tracting the  square  root  of  a  surd,  periods  of  decimal  ciphers  may  he  an- 
nexed, and  the  work  extended  to  any  required  degree  of  exactness. 

3.  If  the  right-hand  decimal  period  contains  hut  one  figure,  annex  a 
decimal  cipher. 

4.  To  extract  the  square  root  of  a  mixed  fractional  number,  first  reduce 
it  to  a  mixed  decimal  number,  or  to  an  improper  fraction. 

PItOBl^:EM8. 

1.  Extract  the  square  root  of  5,476.  74. 

2.  Find  the  value  of  V75.69.  8.7. 

3.  What  is  the  square  root  of  .0389  ? 

4.  A  square  plat  of  ground  contains  87,616  square  feet.  What 
is  the  length  of  one  side  ? 


5.  V881,731  =  what  number  ? 

6.  Extract  the  square  root  of  .455625. 

7.  What  is  the  square  root  of  50,808,384  ?  7, 128. 

8.  The  area  of  a  square  platform  is  1,387^  sq.  ft.     What  is  the 
length  of  one  side?                                               '  37.25  ft. 

9.  Fmd  the  value  of  V.000169.  .013. 

10.  What  is  the  square  root  of  the  fraction  ff^  ? 

11.  V||i  =  what  number?  U- 

12.  The  entire  area  of  the  six  faces  of  a  cubic  block  is  130f  sq. 
in.    What  is  one  dimension  of  the  block  ?  -4f  in. 

13.  Find  the  square  root  of  914|^.  P/^-. 

14.  What  is  the  value  of  Vl5  ?  3.872  -f . 

15.  Extract  the  square  root  of  99.  9.9498  +  . 

16.  Vil27.750734r=what  number?  33.582. 


306  EVOLUTION. 

SECTION  III. 
JEXT^ACTiojv  oj^  cu:bb  ^oot. 

567.  If  we  cube  an  integral  unit  of  each  of  the  first  four 
orders,  we  have 
1«=1         10^=1,000         100^=1,000,000         1,000='=1,000,000,000 

Since  the  cube  of  1  is  1,  and  the  cube  of  10  is  1,000,  the 
cube  of  any  number  between  1  and  10  must  be  a  number 
between  1  and  1,000; 

Since  the  cube  of  10  is  1,000,  and  the  cube  of  100  is 
1,000,000,  the  cube  of  any  number  between  10  and  100 
must  be  a  number  between  1,000,  and  1,000,000; 

Since  the  cube  of  100  is  1,000,000,  and  the  cube  of  1,000 
is  1,000,000,000,  the  cube  of  any  number  between  100  and 
1,000  must  be  a  number  between  1,000,000  and  1,000,000,000; 
and  so  on.     That  is. 
The  cube  of  a  number  expressed  by  •  consists  of 

One  figure  one,  two,  or  three  figures ; 

Two  figures  four,  five,  or  six  " 

Three    "  seven,  eight  or  nine    " 

and  so  on.    Hence 

I.  The  cube  of  any  number  consists  of  three  times  as  many 
figures  as  the  number^  or  one  or  two  less. 

II.  If  a  number  be  separated  into  periods  of  three  figures 
eachj  beginning  with  ones,  its  cube  root  will  consist  of  as  many 
figures  as  there  are  full  and  partial  periods  in  the  number. 

568 1  If  we  write  any  digits,  as  2  and  9,  successively  as 
ones,  tens,  hundreds,  and  so  on,  and  cube  them,  we  have 
2^=     8  20^=     8,000  200^=     8,000,000 

9'=729  90^=729,000  900^=729,000,000 

Examining  these  numbers,  we  see  that 

The  cube  of  the  ones         is  in  the  first      period ; 
"        "      tens  "        second       " 

"        "      hundreds      "        third         " 
and  so  on.    Hence, 


EXTRACTION    OF    CUBE    ROOT.  307 

The  cube  of  the  left-hand  Jigure  of  a  root  is  ivholly  in  the  left- 
hand  period  of  the  power. 

569.  If  we  cube  any  numbers  expressed  by  two  figures, 
as  20  and  25,  60  and  63,  90  and  99,  we  shall  have 

20^=  8,000        60^=316,000        90^=729,000 
25^=15,625         63^=313,047         99=^=970,299 

Comparing  these  roots  and  their  cubes,  we  see  that 

8  13  the  greatest  cube  in  15,  the  thousands  of    15,625 ; 
216      "  "  313,  "  "  "    313,047; 

729      "  "  970,  "  "  "    970,299.    Hence, 

The  greatest  cube  in  the  left-hand  period  of  a  number  is  the 
cube  of  the  left-hand  Jigure  of  the  root. 

570.  We  will  now  cube  the  number  45,  for  the  purpose  of 
seeing  of  what  parts  the  cube  is  composed.  45=40  +  5,  and 
45'=:40  +  5  multiphed  by  40 +  5  multiplied  by  40  +  5. 

FIEST  SOLUTIOK.  SECOND  SOLUTION. 

40     +  5  =      4^ 

(4-0  X  5)  +  5'  =      .  225 
40^  +         (40  X  S)  =     180 

lo""  +  2x{JfO  X  5)  +  5'  =     2025 
Jf.0     +  5  =  Jf5 


{40''  X  5)-\-  2x(40  X  5')  +  5'  =  10125 
40^  ■\-2x{JfO''  X  5)+         {40  X  5')  =  8100 

40'  +  3x{40''  X  5)-\-  3x(40  X  5')  +5'=  91125 

The  several  parts  of  the  final  product,  reading  from  the 

left,  are 

ls#.  The  cube  of  the  tens,  6^,000 

2d.  Three  times  the  square  of  the  tens  x  the  ones,         24,000 
Sd.  Three  times  the  tens  x  the  square  of  the  ones,  3^000 

Uli.  The  cube  of  the  ones,  125 

Thatis,  45^  =  5i,i^5 
The  cube  of  a  number  consisting  of  two  figures,  is  equal  to  the 

cube  of  the  tens,  plus  three  times  the  square  of  the  tens  multiplied 


308  EVOLUTION. 

hy  the  ones,  plus  three  times  the  tens  multiplied  by  the  square  of 
the  ones,  plus  the  cube  of  the  ones. 

571.  Ex.  1.  What  is  the  cube  root  of  91,125? 
Explanation. —  solution, 

4  5  Root. 


Separating      the                     91-125 
mimber    into    pe-  ^  -^    


riods  of  three  fig-       i>^idend.  2712  5 
nres  each,  we  find  ^  ^  i^5 


Jf.8  0  0  Trial  divisor. 

600 

25 


5  Jf.2  5  Complete  divisor. 


that  the  cube  root 

will  consist  of  two  figures  (567,  II.). 

Since  64  is  the  greatest  cube  in  the  left-hand  period,  91, 
we  write  4,  its  cube  root,  for  the  first  figure  of  the  root  (569). 

Taking  64,  the  gTeatest  cube,  from  the  left-hand  period^ 
and  annexing  125,  the  next  period,  to  the  remainder,  we 
have  27,125.  This  number  is  made  up  of  3  times  the  square 
of  the  tens  x  the  ones,  plus  3  times  the  tens  x  the  square 
of  the  ones,  plus  the  cube  of  the  ones  (570)  ;  i.  e.,  of  3  x  40^* 
X  the  ones  +  3  x  40  x  the  square  of  the  ones  +  the  cube 
of  the  ones. 

CaUing  the  first  figure  of  the  root  tens,  and  multiplying 
its  square  by  3,  we  have  4,800  for  a  trial  divisor.  Dividing 
the  dividend,  27,125,  by  the  trial  divisor,  we  obtain  5  for 
the  second  figure,  or  ones,  of  the  root. 

Since  4,800=3  times  the  square  of  4  tens,  and  27,125  = 
3  times  the  square  of  4  tens  x  the  ones,  plus  3  times  4  tens 
X  the  square  of  the  ones,  plus  the  cube  of  the  ones,  we 
add  to  4,800,  the  trial  divisor,  600  (  =  3x4  tens  or  40 
X  the  ones),  and  also  25,  the  square  of  the  ones,  making 
5,425,  the  complete  divisor.  Then,  multiplying  this  com- 
plete divisor  by  5,  the  second  figure  of  the  root,  we  obtain 
27,125,  which  is  made  up  of,  1st,  5  x  5  x  5,  or  the  cube  of 
the  ones;  M,  3  x  40  x  5  x  5,  or  3  times  the  tens  x  the  square 
of  the  ones;  and  Sd,  3  x  40  x  40  x  5,  or  3  times  the  square 
of  the  tens  x  the  c^nes.  We  have  now  used  all  of  the  given 
number,  and  45  is  the  cube  root  required. 


EXTRACTION    OF    CUBE    ROOT.  309 

Ex.  2.  Extract  the  cube  root  of  9,663,597. 
Explanation.  — 


Since    only  two   pe-  ^°^ 

riods  of  figures  are      ^'^^^-^9  7 

considered   at  once,       

in  obtaining  any  fig-       ^^^^^ 
ure  of  the  root,  after 


213 

1200  +  60  +  1 
1261 

132300  +  1890  +  9 
134199 

the  firsWe   regard         |g|^^^ 

the  ngure  or  figures       —-^ 

of  the   root  already 

found  as  tens,  and  the  figure  sought  as  ones;  and  proceed 

in  the  same  manner  as  in  obtaining  the  second  figure. 

572.  Upon  the  principles  in  567-570  is  based  the 
:%zg^e  for   JEJxtractlon    of  Cube    "Eoot, 
I.  To  determine  the  number  of  figures  in  the  root. 
Separate  the  number  into  periods  of  three  figures   each, 
counting  left  and  right  from  ones. 

n.  For  the  first  figure  of  the  root. 

1.  Find  the  root  of  the  greatest  cube  in  the  left-hand  period. 

2.  Subtract  its  cube  from  the  period,  and  to  the  remainder 
annex  the  next  period,  for  a  dividend. 

m.  For  the  second  figure  of  the  root. 

1.  Considering  the  7'oot  already  found  as  tens,  multiply  its 
square  by  3,  for  a  tr^al  divisor,  by  which  divide  the  dividend, 
and  write  the  residtfor  the  second  figure  of  the  root. 

2.  Add  to  the  trial  divisor  3  times  the  product  of  the  tens 
and  ones  of  the  root  already  found,  and  also  the  square  of  the 
ones,  for' a  complete  divisor. 

3.  Multiply  the  complete  divisor  by  the  last  figure  of  the 
root ;  subtract  the  product  from  the  dividend ;  and  to  the 
remainder  annex  the  next  period  for  a  new  dividend. 

rV".  For  each  succeeding  figure  of  the  root. 
Consider  that  part  of  the  root  already  found  as  tens,  and 
proceed  in  the  same  manner  as  in  finding  the  second  figure. 


310  EVOLUTION. 

Notes.— 1.  If  any  dividend  is  less  than  the  divisor,  annex  a  cipher  to  the 
root;  two  ciphers  to  the  trial  divisor,  for  a  new  divisor;  and  the  next 
period  to  the  dividend,  for  a  new  dividend. 

2.  In  extracting  the  cube  root  of  a  surd,  periods  of  decimal  ciphers  may- 
be annexed,  and  the  worlc  extended  to  any  required  degree  of  exactness. 

3.  If  a  right-hand  decimal  period  contains  less  than  three  figures,  supply 
the  deficiency  by  annexing  a  decimal  cipher  or  ciphers. 

4.  If  the  given  number  is  a  fraction,  take  the  cube  root  of  the  numerator 
and  denominator  separately ;  and  if  it  is  a  mixed  fractional  number,  first 
reduce  it  to  an  improper  fraction,  or  to  a  mixed  decimal  number. 

5.  Since  the  trial  divisor  is  less  than  the  true  divisor,  in  obtaining  the  root 
figure  we  must  malie  allowance  for  this  difierence.    See  Manual. 

PJJ  O  B  JL  JEJ  M  S. 

1.  What  is  the  cube  root  of  103,823 ;  and  of  24,389  ?      47;  SO, 


3.  V274.625  =  what  number  ?  6.5. 

3.  .000729  is  the  cube  of  what  number  ? 

4.  What  is  the  length  of  one  side  of  a  cubical  block  that  contains 
2  cu.  ft.  1,457  cu.  in.  ?  1  ft.  5  in. 

5.  Find  one  of  the  three  equal  factors  of  10,218,313.  217. 

6.  Vl31,09~6,512  =  what  number? 

7.  The  length  of  a  square  stick  of  timber,  which  contains  13|^ 
cubic  feet,  is  32  times  its  width  or  thickness.  What  are  its 
dimensions  ? 

Note  6. — If  the  stick  were  cut,  crosswise,  into  33  equal  parts,  each  part 
would  be  a  cube.  2 4  ft.  long,  and  9  in.  square. 

8.  In  digging  a  cellar,  the  length  of  which  was  4  times,  and  the 
width  6  times  its  depth,  192  cubic  yards  of  earth  were  removed. 
What  were  the  dimensions  of  the  cellar  ?  6,  2^,  and  36  ft. 

9.  Extract  the  cube  root  of  187,149.248. 

10.  Of  what  number  is  118,805,247,296  the  cube  ?  4,916. 

11.  In  a  granary  is  a  bin  that  holds  270  bushels.  Its  length  is  3 
times,  and  its  width  If  times  its  depth.     What  are  its  dimensions  ? 

12.  Extract  the  cube  root  of -g^f^,  and  ^V- 

13.  VlGyV^T  and  V4^^  what  numbers  ?  2fj,  and  1.6. 

14.  What  must  be  the  interior  measurement  of  a  side  of  each  of 
two  boxes,  one  of  which  will  hold  a  bushel  of  grain,  and  the  othev 
a  gallon  of  oil  ?  12.907+  in.,  and  6.135+  in. 


SECTION  I.    ■ 

3)  ^I^IJVI  TZOJVS, 

573.  A  Series  is  a  succession  of  numbers  increasing  or 
decreasing,  either  by  a  common  difference  or  by  a  common 
ratio  ;  as  3,  7,  11,  15  ;  and  2,  6,  18,  54. 

Note.— The  numbers  that  form  a  Series  are  the  Terms.  The  first  and  last 
terms  are  thQ  Extre7nes  ;  and  the  other  terms  are  the  Means.  (See  539, 530.) 

574.  An  Ascending  Series  is  one  in  wHch  tbe  terms 
increase  in  regular  order,  from  the  first. 

575.  A  !Descending  Series  is  one  in  which  the  terms 
decrease  in  regular  order,  from  the  first. 

576.  An  oirit?imeHcal  Progression  is  a  series  whose 
terms  increase  or  decrease  by  a  common  difference  ;  as  2, 
7,  12,  17  ;  and  24,  21,  18,  15. 

577.  A  Geometrical  Progression  is  a  series  whose 
terms  increase  or  decrease  by  a  common  ratio ;  as  2, 10,  50, 
250  ;  and  48,  24,  12,  6. 


SECTION  II. 

^  ^ITMM^ETICA.  Z    Z>Zl  O  G  ZiBS  SZOJV. 
578.  In  the  ascending  arithmetical  series  2,  5,  8,  11,  14, 
the  common  difference  is  3,  and  the  terms  are  formed  as 
follows  : 

1st  term,    2 ; 

2d      "        5  =  2  +  3,  or  1st  term  +  common  difference » 

3d      "        8=2  +  3  +  3,  "         "     +  2  times  com.  diff. ; 

4th     "      11  =  2  +  3  +  3  +  3,  "         "     +3      "  " 

6th     "      14=2+3  +  3  +  3  +  3,    "         ''     +4     "  " 

and  the  sum  of  the  series  is  2  +  5  +  8  +  11  + 14=40. 


312  PROGRESSIONS. 

From  this  illustration  we  see,  that  in  any  arithmetical 
series  there  are  five  things  to  be  considered  :  viz.,  the  First 
Term,  the  Last  Term,  the  Common  Difference,  the  Number 
of  Terms,  and  the  Sum  of  the  Series. 

Dividing  40,  the  sum  of  the  series  2,  5,  8,  11,  14,  by  5, 
the  number  of  terms,  we  have  8,  which  is  the  average  of 
all  the  terms,  or  the  Average  Term ;  and  adding  2  and  14, 
the  extremes,  we  have  16,  which  is  two  times  the  average 
term. 

579.  From  these  illustrations  we  deduce  the  following 

Principles  of  A.rithmeUcal  :Pro(/ressio7i, 

I.  Any  term  in  an  ascending  series  is  equal  to  the  first  term, 
plus  the  product  of  the  common  difference  multiplied  by  the 
number  of  the  term  less  1. 

n.  Tlie  difference  of  the  extremes  is  equal  to  the  product  of 
the  common  difference  multiplied  by  the  number  of  terms  less  1. 

in.  The  sum  of  the  extremes  is  equal  to  two  times  the  average 
term  of  the  series. 

JPJJ  OBZE3IS. 

1.  The  first  term  of 

T  'ii  SOLUTION. 

an  ascending  antli- 

metical  series  is  6,  57 -1=56 'times  com.  diff.  is  added. 

the  common  differ-  ^^  x  S=168,  sum  of  additions  to  1st  term. 

ence  is  3,  and  the  6+ 168= 174,  last  term.     (Seel.) 

number  of  terms  is  ^^^ 

57.      What    is    the  6 +{56  x  3)= m,  last  term. 

last  term  ? 

2.  The  first  term  of 

,  -J.  .,,  SOLUTION. 

a  descending  antn- 

metical  series  is  206,  21—1=20,  times  com.  diff.  is  subtracted. 

the  common  differ-  20  xl0=200,sumof  subtractions  from  1st  term. 

ence  is  10,  and  the  206-200=6,  last  term.     (See  II.) 

number  of  terms  is  o>\ 

21.      What    is    the  206-{20xl0)=6,  last  term. 

last  term  ? 


ARITHMETICAL    mOGRESSlON, 


313 


3.  The  first  term  is 
5,  tiie  last  term  is 
117,  and  the  nmiiber 
oftermsislS.  What 
is  the  common  dif- 
ference ? 

4.  The  extremes 
are  7  and  95,  and 
the  common  diflfer- 
ence  is  4.  Find  the 
number  of  terms  in 
the  series. 

5.  The  extremes 
are  3  and  25,  and 
the  number  of  terms 
is  12.  What  is  the 
sum  of  the  series  ? 


SOLUTION. 

117—5=112,  sum  of  additions  to  1st  term. 
15  —  1= IJf,  number  of  additions. 
112-^14=S,  common  difference.     (See  II.) 

Or 

117-5         _  ,.„ 

=  8.  common  difference. 

15-1 

SOLUTION. 

95—7=88,  sum  of  additions  to  1st  term. 
88^4=22,  number      "  " 

22  + 1=23,  number  of  terms. 

Or 

95—7 
[-1=23,  member  of  terms. 

4 

SOLUTION. 

3-^25=28,2  times  the  average  term.   (See  III.) 
28^2=14,  tJie  average  term. 
14x  12=  168,  sum  of  the  series. 

Or 

3  -\-25 
X  12=168,  sum  of  the  series. 


580.  Upon  the  principles  and  examples  in  578,  579,  are 
based  the 

littles  fo7'  Computations  In  Arithmetical  (Progression, 

I.  To  find  either  extreme. 

Multiply  the  common  difference  by  the  number  of  terms  less  1 ; 

and  add  the  product  to  the  less  extreme,  or  subtract  it  from  the 

greater. 

n.  To  find  the  common  difference. 

Divide  the  difference  of  the  extremes  by  the  number  of  terms 
less  1. 

m.  To  find  the  number  of  terms. 

Divide  the  difference  of  the  extremes  by  the  common  differ- 
ence,  and  add  1  to  the  quotient. 

rV.  To  find  the  sum  of  the  series. 
Multiply  one  half  the  sum  of  the  extremes  by  the  number  of 

terms.  see  ManuaL 

14 


314  PROGRESSIONS. 

l^Ji  OBZJ^MS, 

6.  The  less  extreme  of  an  arithmetical  series  is  5,  the  common 
diflference  is  7,  and  the  number  of  terms  is  13.  What  is  the  greater 
extreme  ?  89. 

7.  Find  the  greater  extreme  of  the  progression  of  which  19  is  the 
less  extreme,  3  is  the  common  difference,  and  57  is  the  number  of 
terms. 

8.  A  boy  14  years  old  was  apprenticed  to  a  trade,  and  was  to  re- 
ceive $50  the  first  year,  and  an  increase  of  $75  yearly,  till  he  was  of 
age.    How  much  did  he  receive  the  last  year  ?  $500. 

9.  The  greater  extreme  of  an  arithmetical  series  is  215,  the  com- 
mon difference  is  13,  and  the  number  of  terms  is  15.  What  is  the 
less  extreme  ?  33. 

10.  A  man  who  owns  a  plot  of  18  building  lots,  asks  $1,000  for  the 
one  nearest  the  city,  and  $20  less  for  each  succeeding  lot.  What  is 
his  price  for  the  lot  farthest  from  the  city  ? 

11.  The  extremes  of  a  series  of  60  terms  are  13  and  249.  What  is 
the  common  difference  ?  ^. 

12.  If  a  laborer  has  $16  deposited  in  a  savings-bank  on  Jan.  1,  and 
$484  on  Dec.  30  following,  what  are  his  average  weekly  deposits  ? 

13.  The  extremes  are  4|  and  67f,  and  the  common  difference  is  If. 
What  is  the  number  of  terms  ?  ^6. 

14.  In  how  many  years  will  the  value  of  a  piece  of  property  be 
doubled,  if  it  increases  in  value  16;^  the  first  year,  and  7fo  each  suc- 
ceeding year  ? 

15.  What  is  the  sum  of  the  natural  series  of  numbers  1,  2,  3,  4, 
and  so  on  to  1,000  inclusive  ?  500,500. 

16.  What  is  the  number  of  strokes  made  by  a  clock  in  12  hours  ? 

17.  The  less  extreme  is  |^,  the  common  difference  is  |^,  and  the 
number  of  terms  is  50.     What  is  the  greater  extreme  ? 

18.  If  you  deposit  $25  in  a  savings-bank  the  first  week  of  the 
year,  and  $5  each  succeeding  week,  how  much  will  you  deposit  in 
the  year  ?  $280. 

19.  What  is  the  84th  term  of  the  series  90f ,  90,  89^,  etc.  ? 

30.  Insert  32  arithmetical  means  between  the  extremes  13  and 
244.  1st  mmn,  20;  S2d  mean^  237. 


GEOMETKICAL    PROGRESSION.  315 

21.  If  the  water  in  a  lake  is  16,}  feet  deep  1  rod  from  a  pier,  and 
the  bottom  has  a  uniform  slope  of  S^-  feet  to  the  rod,  at  wliat 
distance  from  the  pier  is  the  water  300  feet  deep  ?  82  rods. 

32.  Find  the  sum  of  100  terms  of  the  series  19^'^,  19|-,  20tV,  etc. 


SECTION    III. 

GJSOMBT'RIC;^!,     'P  ^O  G'B  ^EJS  S 10  J^, 

581.  In  the  ascending  geometrical  series  %  6,  18,  54, 162, 
the  ratio  is  3,  and  the  terms  are  formed  as  follows  : 


1st 

term,    2 ; 

2d 

"        6=2x3,               or  1st  term  x  ratio, 

2  X  3'; 

3d 

"      18  =  2x3x3,              "        "      X  square  of  ratio, 

2  X  3^; 

4th 

"      54=2x3x3x3,        "        "      x  cube            "" 

2  X  3^; 

5th 

"    162=2x3x3x3x3,'-        "      x  4th  power,  " 
and  the  sum  of  the  series  is  2  +  6  +  18  +  54  +  162=242 

2  X  3*; 

From  this  illustration  we  see  that,  in  any  geometrical 
series  there  are  five  things  to  be  considered  :  viz.,  the  Fir^t 
Term,  the  La%t  Term,  the  Ratio,  the  Number  of  Terms,  and 
the  Sum  of  the  Series. 

If  we  take  the  above  series,  2,  6,  18,  54,  162,  multiply  it 
by  the  ratio,  3,  placing  the  terms  of  the  products  over  the 
corresponding  numbers  of  the  series,  and  then  subtract  the 
series  from  the  product  (or  3  times  the  series),  the  terms 
consisting  of  like  numbers  will  disappear,  and  we  shall  have 

3  times  the  series,  6        18        54        162        486 

Series,       2        6        18        54        162 

(8  times— 1  time=)2  times  the  series=486 — 2=484 

Since  484  is  2  times  the  series,  484 +-2  ==242  is  1  time  the 

series,  or  the  sum  of  the  series. 

Note.— The  pupil  will  notice  that  2  is  the  first  term  of  the  scries,  4S6  b 
8  times  the  last  term,  and  the  divisor,  3,  is  the  ratio  less  1. 


316 


PROGRESSIONS. 


582.  From  these  illustrations  we  deduce  the  following 

Principles  of  Geometrical  Progress  ion. 

I.  The  first  term  and  the  ratio  are  the  only  factors  used  in 
forming  a  series. 

II.  In  any  term  of  a  series,  the  first  term  is  a  factor  once. 

m.  In  any  term  of  an  ascending  series,  the  ratio  is  a  factor 
as  many  times  as  the  number  of  terms  less  1. 

rV.  The  number  of  factors  used  informing  any  term,  is 
equal  to  the  number  of  the  term. 

V.  The  product  of  the  ratio  and  the  greater  extreme  of  a 
series,  ^ninus  the  less  extreme,  is  as  many  times  the  sum  of  the 
series,  as  is  expressed  by  the  ratio  less  1. 


1.  The  first  term  of 
an  ascending  geo- 
metrical series  is  4, 
the  ratio  is  3,  and  the 
number  of  terms  is 
7.  What  is  the  last 
term? 

2.  The  first  term 
of  a  descending  geo- 
metrical series  is  96, 
the  ratio  is  2,  and 
the  number  of  terms 
is  6.  What  is  the 
last  term  ? 

3.  The  first  term  of 
a  geometrical  pro- 
gression is  7,  the 
last  term  is  567,  and 
the  number  of  terms 
is  5.  What  is  the 
ratio  ? 


JPJiOBI^JE3IS, 

SOLXTTION. 

7—1=6,  times  the  ratio  is  a  factor.    (See  in.) 
^^=64,  product  of  ratio  used  as  a  factor. 
4  X  64=256,  last  term.     (See  I.) 

Or 

4  X  S''-^ =4  X  2^' ="256.  last  term. 

60LUTI0X. 

6—1=5,  times  tlie  ratio  is  a  dimsar.  (See  III.) 
2^=32,  product  of  ratio  used  as  a  divisor. 
96^32=3,  last  term. 
Or 
^6 -^2'-' =96 -^2" =3,  last  term. 


567-7-7=81,  product  of  ratio  used  as  a  factor. 
5—1=4,  times  ratio  is  a  factor.    (See  III.) 
V5J  =3,  ratio.      See  Manual,  Reference  310. 
Or 

^'1/567       W567      ^      ,. 


GEOMETRICAL    PROGRESSION.  317 

4.  The  extremes  of  solution. 

a    geometrical  pro-  1536-^6=256^  prod,  of  ratio  used  as  a  factor. 

gression   are  6  and  256-^4=64,  64^4=16, 16-^4=4,  4-^4=1. 

1  536  and  the  ratio  -^  ('^^'^  times  the  ratio  is  a  factor)  +  1  (the  time 
is  4     What  is  the  ^^^^  ^^^^  extreme  is  a  factor) =5^  number  of 

number  of  terms  ?  ^^^^^«-     (See  IV.) 

SOLUTION'. 

5.  The  first  term        ^^^  ^  5=1875,  5  times  the  last  term. 

IS  3,  the  last  term  is  ^sjj^_s^  ^g^^,  4  times  the  series.    (See  V.) 

375,  and  the  ratio  is  ^s72-^4^  468,  sum  of  the  series. 

5.    What  IS  the  sum  '  q^ 

of  the  series  ?  ?11^^  =468,  sum  of  the  series. 

583t  Upon  the  principles  and  examples  in  581,  582,  are 
based  the 

(Rules  for  Computations  in   Geometrical  "Progress ion, 

I.  To  find  the  greater  extreme. 
Raise  the  ratio  to  a  power  1  less  than  the  number  of  terms, 
and  multiply  the  result  by  the  less  extreme. 

n.  To  find  the  less  extreme.  . 
Baise  the  ratio  to  a  power  1  less  than  the  number  of  terms, 
and  divide  the  greater  extreme  by  this  result. 

III.  To  find  the  ratio. 
Divide  the  greater  extreme  by  the  less,  and  extract  that  root 
of  the  quotient  whose  index  is  1  less  than  the  number  of  terms. 

TV.  To  find  the  number  of  terms. 
Divide  the  greater  extreme  by  the  less,  and  this  and  each  suc- 
ceeding result  by  the  ratio,  till  the  quotient  is  1. 

The  number  of  divisions  will  be  the  number  of  terms. 

V.  To  find  the  sum  of  the  series. 
Multiply  the  last  term  by  the  ratio,  from  the  product  subtract 
ih?  first  term,  and  divide  the  remainder  by  the  ratio  less  1. 

See  ManuaL 


318  PBOGKESSIONS. 

6.  The  first  term  of  an  ascending  geometrical  series  is  7,  and  the 
ratio  is  3.     What  is  the  6th  term  ?  1,701. 

7.  The  less  extreme  is  13,  the  ratio  is  4,  and  the  number  of  terms 
is  7.    What  is  the  greater  extreme  ? 

8.  The  5th  term  of  an  ascending  series  is  5,625,  and  the  ratio  is  5. 
What  is  the  iirst  term  ?  p. 

9.  The  greater  extreme  is  845,824,  the  ratio  is  2,  and  the  number 
of  terms  is  12.     What  is  the  less  extreme  ? 

10.  The  extremes  of  a  progression  of  5  terms  are  4  and  64.  What 
is  tlie  ratio  ?  2. 

11.  The  four  terms  of  a  proportion  are  in  geometrical  progression, 
and  the  extremes  are  8  and  2,744.     What  is  the  proportion  ? 

Note  1.— First  find  the  ratio.  8  :  56  :  :  S92  :  2744. 

12.  The  extremes  of  a  series  are  3  and  234,375,  and  the  ratio  is  5. 
How  many  terms  are  there  in  the  series  ?  8. 

13.  The  extremes  are  1  and  y^,  and  the  ratio  is  -^.  Find  the 
number  of  terms. 

14.  The  extremes  of  a  series  are  2  and  4,374,  and  the  ratio  is  3. 
What  is  the  sum  of  the  series  ?  .   6^560. 

15.  What  is  the  sum  of  9  terms  of  the  series  2,  10,  50,  etc.  ? 
Note  2. — The  greater  extreme  must  first  be  found.  976 j  562. 

16.  What  debt  can  be  discharged  in  a  year,  by  paying  1  cent  the 
first  month,  3  cents  the  second,  9  cents  the  third,  and  so  on,  in  that 
ratio,  for  the  12  months?  $2,657.20. 

17.  If  it  were  possible  for  a  person  having  only  1  cent,  ip  double 
his  money  every  month  for  4  years,  how  much  money  would  he 
have  ?  $2,814,749,767,106.56. 

18.  What  is  the  sum  of  the  series  18  +  6  +  2  +  f  +  f,  and  so  on, 
to  infinity  ? 

Note  3.— When  the  number  of  terms  in  a  descending  geometrical  series 
is  infinite,  the  series  is  called  an  Infinite  Series,  and  the  last  term  is  0. 


3-1        —  "^ 

19.  Find    the    sum    of    the    infinite   series    100  +  25  +  61  +  1^? 
etc.  133^. 

20.  What  is  the  sum  of  the  series  1,  |,  -|-,  \,    .     .     .     to  0  ? 


INTEEEST    BY    P  HOG  RES  SIGNS,  319 

SECTION    IV. 

IjYTBOiBS  2'   :sr   T^OG^BSSIOJSrS. 

584.  In  Simple  Interest,  the  amount  of  any  sum  is  equal  to 
the  lorincipal,  plus  the  product  of  the  interest  for  1  year 
multipHed  by  the  time  in  years  (see  461)  ;  the  principal  and 
the  amount  due  at  the  close  of  the  first  year  are  both  taken 
into  account ;  and  the  amounts  due  at  the  end  of  the  several 
years  form  an  arithmetical  series.   Hence,  in  computations  of 

Shnple  J?ite7'est  by  A.rlthnietlcal   Progression, 

I.  Tlie  principal  is  the  less  extreme  of  an  arithmetical  series  ; 
H.   TJie  interest  for  1  year  is  the  common  difference  ; 
in.   TJie  number  of  years  plus  1  is  the  number  of  terms  ;  and 
lY.   TJie  amount  is  the  greater  extreme. 

585.  In  Compound  Interest,  the  amount  of  any  sum^  for 
any  year,  is  found  by  multiplying  the  amount  due  at  the 
end  of  the  preceding  year  by  1  plus  the  rate  %  (see  4G2); 
and  the  amounts  due  at  the  end  of  the  several  successive 
years  form  a  geometrical  series.    Hence,  in  computations  of 

Cot7?pound  Interest  by  Geojnetrical  Progression, 

I.  l%e  principal  is  the  less  extreme  of  a  geometrical  senes.; 

II.  1  plus  the  rate  is  the  ratio  ; 

III.  The  number  of  years  plus  1  is  the  number  of  terms  ;  and 

IV.  Tlie  amount  is  the  greater  extreme. 

PROBLEMS. 

1.  What  is  the  amount  of  $500  for  8  years,  at  7^  ?  at  Qfo  'i 

2.  I  have  a  lease  of  a  building  for  9  years,  at  $50  a  year.  If  I 
allow  Qfo  interest  on  the  rents  from  the  time  they  are  due,  and  pay 
the  whole  amount  at  the  expiration  of  the  lease,  how  much  must  I 
then  pay  ? 

Note  1. — As  there  is  no  principal  at  the  commencement  of  the  first 
year,  there  are  only  9  terms.  $558. 

3.  If  I  save  $150  each  year,  and  put  it  at  interest  at  10^,  how 
much  will  my  savings  amount  to  m  10  yoars?  $2,175. 


320  PllOGRESSIONS. 

4.  If  a  soldier's  pension  of  $100  per  annum  remains  unpaid  for 
9  years,  how  much  will  then  be  due  him,  allowing  Gfo  simple 
interest  ? 

Note  2.— Since  the  first  year's  pension  lias  been  clue  9  years,  and  the  last 
year's  pension  is  now  due,  there  are  10  terms  in  the  series.  $1,270. 

5.  What  is  the  present  worth  of  an  annuity,  or  annual  income, 
of  $300  having  6  years  to  run,  money  being  worth  Sfo  ? 

Note  3. — The  amount  of  the  annuity  due  at  the  end  of  the  0  years,  is  the 
sum  to  be  discounted,  at  ^%. 

6.  I  hired  a  mill  for  5  years,  at  an  annual  rent  of  $600,  and  paid 
the  rent  in  advance,  less  the  discount  of  6^.     How  much  did  I  pay  ? 

7.  After  three  years,  I  shall  come  into  possession  of  property  that 
pays  $400  annually.  How  much  ready  money  can  I  borrow,  by  hy- 
pothecating 5  years  of  this  income,  and  allowing  10^  for  the  loan  ? 

Note  4. — The  sum  due  in  5  years  after  the  income  commences,  or  8  j^ears 
from  the  present  time,  is  the  amount  to  be  discounted  for  8  years,  at  10^. 

8.  What  is  the  present  worth  of  an  annuity  4  years  in  reversion 
(^.  e.  to  commence  in  4  years),  and  then  having  7  years  to  run, 
money  being  worth  8^  ? 

9.  What  is  the  present  worth  of  a  perpetual  annuity  of  $1,500, 
to  commence  7  years  hence,  if  discounted  at  6^  ? 

Note  5. — The  annuity  is  the  interest  of  a  principal  that  will  earn  $1,500, 
at  &%, 

10.  How  much  will  $10,000  amount  to  in  5  years,  at  6^  compound 
interest  ?  $13,382.26. 

11.  What  is  the  compound  interest  of  $425  for  5  years,  at  7^^  ? 
13.  If  you  dci3osit$500  in  a  savings-bank  that  pays  5'^  on  deposits, 

compound  interest  payable  quarterly,  how  much  will  your  money 
amount  to  in  3  years  ? 

13.  What  sum  of  money,  at  6^  compound  interest,  will  amount 
to  $89.54  in  4  years  ?  $70.92. 

14.  The  amount  is  $33,153.83,  the  time  is  8  years,  and  the  rate  is 
10^  compound  interest.     What  is  the  principal  ?  $15,000. 

15.  Find  the  amount  of  an  annuity  of  $185  for  4  years,  at  7^ 
compound  interest.  $821.39. 

16.  A  clerk  deposited  $75  in  a  savings-bank  every  6  months, 
upon  which  he  received  Qfo  interest  compounded  semi-annually. 
How  much  was  standing  to  his  credit  at  the  end  of  4  years  ? 

17.  How  much  is  an  annuity  of  $1,300  per  annum  worth  in  10 
years,  at  5^  compound  interest  ?  $15,093.^7. 


SECTION  I. 

586.  Mensieratlon  embraces  the  processes  of  measuring  and 
computing  tlie  length  of  lines,  the  area  of  surfaces,  and  the  capacity 
of  solids  and  spaces.     (See  Chap.  2,  Sec.  VIII.) 

587.  A  Curve  Z,i7ie  continually  changes  its  direction,  no  part  of 
it  being  a  straight  line. 

688.  ^a7'aUel  Zlnes  run  in  the  same  direction, — 

at  the  same  perpendicular  distance  apart.  ^-^ — -^^ 

589.  An  A.ciete  ^7if/le  is  less  than  a  right  /^  \ 
angle;  as  ABG. 

590.  An  Obtuse  A.7igle  is  greater  than  a 
right  angle ;  as  ABB. 

Note. — Acute  and  obtuse  angles  are  Oblique  Angles. 

591.  A  "Plaiie  Fiffm^e^  or  a  ^lane,  is  a  level  surface  bounded 
by  lines. 

592.  A  ^otys^on  is  a  plane  bounded  by  straight  lines. 

593.  A  ^cffiUai''  Toty^on  has  all  its  sides,  and  also  all  its 
angles  equal. 

594.  A  2rla77ffle  is  a  polygon  of  three  sides. 

595.  A  mght-A.7iffled  2ria7iffle  has  one  right 
angle ;  as  ABC. 

596.  A  JIypothe7itise  is  the  longest  side  of  a 
right-angled  triangle ;  as  ^  C. 

Note. — A  triangle  having  three  acute  angles  is  an  Acute- Angled  Triangle ; 
one  having  one  obtuse  angle  is  an  Obtuse-Angled  Triangle  ;  one  having  all 
its  sides  equal  is  an  Equilateral  Triangle ;  one  having  two  sides  equal  is  an 
Isosceles  Triangle ;  and  one  having  all  its  sides  unequal  is  a  Scalene  Triangle, 

597.  A  Quadrilateral  is  a  poly-  ^''^  ^^''""'^^• 
gon  of  four  sides. 

598.  A  ^arallelosrram  is  a 
quadrilateral  whose  opposite  sides  are 
parallel,  .ind  consequently  equal. 


322 


MENSURATION. 


Note.— If  all  the  sides  of  an  oblique-angled  parallelogram  are  equal,  it 
is  a  JRhombus,  or  Bhomb  ;  if  only  the  opposite  sides  are  equal,  it  is  a  Bhoirv- 
bold. 

599.  A  Trapezoid  is  a  quadrilateral  having  only  two  sides 
parallel. 

600.  A  Trapezium  is  a  quadrilateral  having         /  ^~\ 
neither  two  of  its  sides  parallel.                                  /  \ 

601.  A  ^iaffonal  is  a  straight  line  joining    L A 

two  oj)posite  angles  of  a  figure. 

Notes. — 1.  A  regular  polygon  of  five  sides  is  a  Perv- 
tagon;  one  of  six  sides  is  a  Hexagon;  one  of  seven 
sides  is  a  Heptagon ;  one  of  eight  sides  is  an  Octagon ; 
one  of  nine  sides  is  a  Nbnagon  ;  and  one  of  ten  sides  is  a  Decagon. 

3.  Any  polygon  of  more  than  three  sides  may  be  divided,  by  diagonals 
all  meeting  at  one  angle,  into  as  many  triangles  less  2  as  the  polygon  has 
sides. 

3.  The  total  length  of  the  sides  of  a  polygon  is  its  Perimeter  ;  and  the 
length  of  the  circumference  of  a  circle  is  its  Periphery. 

602.  The  Sase  of  a  figure  is  the  side  on  which  it  is  supposed 
to  stand; 

603.  The  Vertex  is  the  point  opposite,  and  furthest  from  the 
base ;  and 

604.  The  A.Uitude  is  the  perpendicular  height  of  \)aQi  vertex 
above  the  base. 


605.  A  l*rism  is  a  solid  whose  bases  or  ends  are  equal,  parallel 
polygons,  and  whose  sides  are  parallelograms. 

608.  A  Cylinder  is  a  solid  whose  bases  or  ends  are  equal,  par- 
allel circles. 

607.  A  ^j/7'amid  is  a  solid  whose  base  is  a  polygon,  and  whose 
sides  are  triangles,  terminating  in  a  point  or  vertex. 

608.  A  Cone  is  a  solid  whose  base  is  a  circle,  and  whose  top  is 
a  point  or  vertex. 


MENSURATION    OF    LINES. 


323 


609.  A  Sp?ie7'e,  or  a  Globe^  is  a  solid  bouuded  by  one  surface, 
whicli,  in  every  jpart,  is  equally  distant  from  a  point  within,  called 
its  center. 

Note.— One  half  of  a  sphere  is  a  Hemisphere. 

610.  Similar  Surfaces  have  their  several  angles  equal  each  to 
each,  and  their  sides  about  the  equal  angles  proportional. 

611.  Simita?^  Solids  are  contained  by  the  same  number  of 
similar  surfaces,  similarly  situated. 


SECTION  II. 

MJBJJV-SZra^ATIOJV    OF    ZIJVBS. 

612.  Some  of  the  principles  of  mensu- 
ration can  only  be  proved  by  a  Geometri- 
cal analysis.  Thus,  this  diagram  illus- 
trates the  first  of  the  following  Geomet- 
rical Principles,  but  the  illustration  is 
not  an  analysis  of  the  principle. 

Geometrical    ^rinci^les. 

I.  The  square  of  the  hypoihenuse  of  a  right- 
angled  ti'iangle  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 

II.  The  diameter  of  a  circle :  the  circumference : :  113  :  355. 

Note.— By  II.  we  find  that,  if  the  diameter  of  a  circle  is  1,  the  circumfer- 
ence is  3.14159,  nearly.  For  ordinary  purposes  it  is  sufficiently  accurate  to 
call  the  circumference  of  a  circle,  3f  times  the  diameter. 

PMOBJOEMS. 

1.  Robert  lives  117  rods  north,  and  the  school-house  is  156  rods  east, 
from  the  corners.  What  is  the  distance  across  the  fields  from  Robert's 
house  to  the  school-house  ?  195  rd. 

2.  What  is  the  length  of  a  hand-rail  to  a  flight  of  16  stairs,  each  12 
inches  wide  and  9  inches  high?  20  ft. 

3.  My  house  is  24  ft.  wide,  the  ridge  is  9  ft.  liigher  than  the  side 
walls,  and  the  eaves  project  1  ft.  6  in.  beyond  the  sides  of  the  house. 
How  wide  is  each  side  of  the  roof?  16  ft.  6  in. 

4.  A  ladder  39  ft.  long  reaches  to  the  top  of  a  building,  when  its 
foot  stands  15  ft.  from  the  building.    How  high  is  the  building  ? 


324 


ENSURATION 


5.  From  the  top  of  a  certain  building,  36  feet  liiffli,  to  the  opposite 
Bide  of  the  street,  is  164  feet.     How  wide  is  the  street  ?  160  ft. 

C.  Two  streets,  one  48  and  the  other  64  feet  wide,  cross  at  right 
angles.    What  is  the  distance  between  the  diagonal  corners  ? 

7.  What  is  the  side  of  a  square  whose  diagonal  is  50  feet? 

8.  Round  a  cylinder  5  ft.  10  in.  high  and  1  ft.  in  circumference,  a 
string  is  wound  spirally  from  bottom  to  top,  passing  14  times  round. 
How  long  is  the  string  ? 

9.  The  slant  lieight  of  a  cone  is  21.8  inches,  and  the  diameter  of  the 
base  is  2.64  inches.     How  high  is  the  cone  ?  21.76  in. 

10.  A  pole  45  ft.  high  is  supported  by  three  guys  attached  to  the 
top,  and  reaching  the  ground  at  the  distances  of  60  ft.,  108  ft.,  and 
200  ft.  from  the  foot  of  the  pole.     What  are  the  lengths  of  the  rods  ? 

75  ft.;   117  ft.;  205ft. 

11.  What  is  the  circumference  of  a  circle  8  feet  in  diameter? 

13.  What  is  the  length  of  the  tire  on  a  carriage  wheel  5  feet  in 
diameter  ? 

13.  What  is  the  circumference  of  a  lake  721  rods  in  diameter  ? 
14   What  is  the  girth  of  an  oak  log  which  is  33  inches  through? 

15.  Find  the  diameter  of  a  circle  which  is  33  rods  in  circumference. 

16.  In  a  park  is  a  fountain  whose  basin  is  3ch.  201.  in  circumfer- 
ence.   What  is  the  diameter  of  the  basin  ? 

17.  The  extreme  end  of  the  minute-hand  of  a  town  clock  moves  for- 
ward 19  inches  in  12  minutes.     How  long  is  the  minute-hand  ? 


SECTION   III. 

MBJVSU^A.TTOJ^     01^    S  17  (R  JFA  C  JS  S, 

613.  By  examining  this  diagram,  we  see 
Ut.  The  diagonal  AG  divides  the  parallel- 
ogram into  two  equal  parts ;  and  conse- 
quently, the  area  of  the  triangle  ABG  is 
equal  to  one  half  the  area  of  the  parallelo- 
gram ABGD,  or  to  \oiAB^  BG.   (See  188). 

2d.  The  areas  of  the  triangles  AFE  and 
BFE  are  equal  to  one  half  the  areas  of  the 
parallelograms  A  FED  and  BFEG^  respectively ;  consequently,  the 


MENSURATION    OF    SURFACES.  325 

area  of  the  whole  triangle  ABE  is  equal  to  one  half  the  area  of  the 
parallelogram  AJBCB,  or  to  the  area  of  the  triangle  ABO.     Hence 

I.  The  area  of  a  triangle  is  equal  to  one  Jialf  the  areu  of  a  parallelo- 
gram liaving  the  same  tase  and  altitude ;  or 

II.  To  one  half  tlie  product  of  its  lase  multiplied  ly  its  altitude. 

614.  If  in  the  rhombus  ABGD,  the  line  DE  be 
drawn  perpendicular  to  the  base,  and  the  part  AED 
be  placed  on  the  opposite  side,  the  line  AD  on  BG, 
the  figure  EFGD  will  be  a  square. 

By  the  same  process,  the  rhomboid  will  be  re-    ^  ^  B  F 

duced  to  a  rectangle.    Hence 

The  area  of  any  parallelogram  is  equal  to  the  'product  of  the  hase 
multiplied  hj  the  altitude. 

615.  If  the  right-angled  triangles  Aae  and  Bha 
be  applied  to  the  spaces  Dde  and  Gce^  respec- 
tively, the  figure  abed  will  be  a  rectangle,  equal 
in  area  to  the  trapezoid  ABGB,  because  the    A  a  b  B 
side  DC  will  be  increased  as  much  as  the  side  J.-B  is  diminished ; 

AT}  I  fn 
AB  +  GD  will  equal  al  +  cd;  and  ah  or  cd  will  equal ^ -.   Hence 

616.  The  area  of  a  trapezoid  is  eqwil  to  the  product  of  one  half  the 
sum  of  its  parallel  sides  multiplied  Tyy  its  altitude. 

617.  From  what  has  been  said  in  613,  614,  it  is  evident  that 

I.  The  area  of  any  polygon  is  equal  to  the  sum  of  the  areas  of  any 
set  of  ti'iangles  into  which  it  may  he  divided  ;  and 

II.  The  area  of  a  regular  polygon  is  equal  to  one  half  the  product  of 
the  periphery  multiplied  lyy  the  altitude  of  one  of  its  equal  triangles. 

618.   Geo77iet7^lcat  'Prhiclples, 

I.  The  area  of  a  circle  is  equal  to  the  product  of  one  half  the  circunrv- 
ference  and  one  half  the  diameter  ;  or 

II.  To  one  fourth  the  product  of  the  circumference  and  diameter ;  or 
ni.  To  the  product  of  the  square  of  the  diameter  multiplied  lyy  .755^. 

IV.  The  surface  of  a  sphere  or  globe  is  equal  to  4  times  the  area  of  a 
circle  of  the  same  diameter, 

V.  The  areas  of  similar  figures  a/re  to  each  other  as  are  the  squares 
of  any  one  of  their  similar  dimensions.    See  Manunl 


^6  MENSUPvATION. 

PItOB  LEMS. 

1.  The  base  of  a  riglit-angled  triangle  is  12  inclies,  and  the  perpen- 
dicular is  8  inches.     What  is  the  area  ?  JfS  sq.  in. 

2.  How  many  feet  of  boards  will  it  take  to  cover  the  gable  of  a  barn 
33  feet  wide,  the  ridge  being  8  feet  above  the  plates  ? 

3.  The  base  of  a  triangle  is  8  ft.  1  in.,  and  its  area  is  2,861^  sq.  in. 
What  is  its  altitude  ?  J^  ft.  11  in. 

4.  The  base  of  a  rhomboid  is  223  feet,  and  its  altitude  is  96  feet. 
What  is  its  area? 

5.  Find  the  area  of  a  trapezoid  whose  sides  are  9  and  17  inches  long, 
and  13  inches  apart,  1  sq.ft.  25  sq.  in. 

6.  How  much  lumber  in  an  inch  board  12  ft,  long,  16  in.  wide  at 
each  end,  and  8  in.  wide  in  the  middle  ?  12  sq.  ft. 

7.  What  is  the  area  of  a  circle  20  feet  in  diameter  ? 

8.  My  horse  is  tied  to  a  stake  in  the  pasture,  by  a  rope  11  ft.  long. 
On  how  much  land  can  he  graze  ?  380.1336  sq.  ft. 

9.  The  area  of  the  bottom  of  a  tin  pan  is  196  sq.  in.  What  is  its 
diameter?  15.79  in. 

10.  How  many  square  inches  of  map  surface  on  a  15-inch  school 
globe?  706.S6. 

11.  The  slant  height  of  a  pyramid  is  11  inches,  and  the  base  is  4 
inches  square.     How  many  square  inches  on  the  entire  surface  ? 

12.  The  periphery  of  the  base  of  a  cone  is  40  inches,  and  the  slant 
height  is  38  inches.  How  many  square  inches  are  there  on  the  lateral 
or  convex  surface  ? 

Note. — The  cone  may  be  regarded  as  a  pyramid  of  an  infinite  number  of 
sides,  and  the  periphery  of  its  base  as  the  sum  of  the  bases  of  all  the  tri- 
angles which  form  its  convex  surface.  5  sq.ft.  40  sq.  in. 

13.  What  is  the  surface  of  a  prism  18  ft.  long  and  21  in.  square  ? 

14.  What  is  the  surface  of  a  round  pillar  14  inches  in  diameter  and 
80  feet  long? 

15.  How  many  feet  of  inch  lumber  in  a  box  6  ft.  6  in.  long,  4  ft.  2  in. 
wide,  and  3  ft.  2  in.  deep,  inside  measurement  ? 

16.  Two  men  start  from  the  same  place,  at  the  same  time.  One  of 
them  travels  south,  at  the  rate  of  3  miles  an  hour,  and  the  other 
west  at  the  rate  of  4  miles  an  hour,  for  7  hours.  They  then  travel 
directly  towards  each  other,  at  the  rate  of  8^  miles  an  hour,  till  they 
meet.  How  many  hours  do  they  travel,  and  how  many  square  miles 
do  they  travel  round  ? 


MENSURATION    OF    CAPACITIES.  327 

SECTION  IV. 

MJSJSrSU^A.TIOJV'    OJP"    CA1>ACITIBS. 

619.  The  capacity  of  a  prism  or  a  cylinder  ^r  7\ 

4,  5,  or  6  feet  in  length,  is  4,  5,  or  6  times  as  >^B|   ,      ji^^^ 

much  as  1  foot  in  length  of  the  same  prism  m||^^^^jj;^^ 

or  cylinder.     (See  191.)     Hence,  ^'m«...JlllLl  i      i  n 

Tlie  capacity  of  a  prism  or  cylinder  is  equal  /f!iii\^^===z:^.:,:^L=A 
to  the  product  of  the  area  of  its  tase  multiplied  P  ^  '^^  "  Sj^^ 
ly  its  length.  \  ^^^^B=— -^^ 

Note.— Lumber  1  incli  thick  or  less  is  sold  by  surflicc  measure.  If  more 
than  1  inch  thick,  it  is  computed  at  this  thickness ;  i.  e.,  the  product  of  the 
surface  measure  in  square  feet  multiplied  by  the  thickness  in  inches,  is  the 
number  of  feet  of  lumber  of  standard  thickness. 

620.   Geo77ietricat  'Prhiciples, 

I.  The  solidity  of  a  pyramid  is  -j-  that  of  a  prism^  and  the  solidity 
of  a  cone  g-  that  of  a  cylinder^  having  the  same  'base  and  altitude. 

II.  The  solidity  of  a  sphere  is  §  that  of  a  cylinder  whose  diameter 
and  altitude  are^  each^  equal  to  the  diameter  of  the  sphere. 

III.  T7ie  capacities  of  similar  solids  are  to  each  other  as  are  the  cubes 
of  any  one  of  their  similar  dimensio7is.    See  Manual. 

PROBLEMS. 

1.  The  ends  of  a  prism  20  feet  long  are  right-angled  triangles,  the 
two  shorter  sides  of  each  of  which  measure  16  and  30  inches.  Find  the 
cubic  contents  of  the  prism. 

3.  How  many  feet  of  timber  in  a  log  31  feet  long  and  17^  inches  in 
diameter?  51.78  +  . 

8.  I  have  a  cylindrical  cistern  6  feet  in  diameter  and -8  feet  deep. 
What  is  its  capacity  in  hogsheads  ?  26hhd.  54-05+ gal. 

4.  The  area  of  the  base  of  an  octagonal  pyramid  is  78  sq.  ft.,  and  its 
altitude  is  19  ft.  6  in.    Wliat  are  its  cubic  contents  ? 

5.  What  are  the  cubic  contents  of  a  cone  7  ft.  in  diameter  at  the 
base,  and  16  ft.  9  in.  high  ? 

6.  Find  the  solidity  of  a  13-inch  school  globe. 

7.  The  capacity  of  a  hollow  globe  of  glass  is  65.45  cubic  inches. 
What  is  its  diameter? 


328  MENSURATION. 

8.  A  leaden  ball  1  inch  in  diameter  weighs  -^^  lb.  How  much,  does 
a  leaden  ball  5  inches  in  diameter  weigh  ?  26-^^  j-  lb. 

9.  A  cast-iron  ball  4  inches  in  diameter  weighs  9  lb.  What  is  the 
weight  of  a  cast-iron  ball  7  inches  in  diameter  ?  ^8^3,  II), 

10.  A  marble  monument  consists  of  a  pedestal  18  inches  square  and 
3  feet  high,  on  which  stands  a  pyramid  16  inches  square  and  7  feet 
high.     What  did  it  cost,  at  $16.25  per  cubic  foot  ?  $177.09. 

11.  A  log  chain  and  3|  quarts  of  water  fill  a  cubical  box  whose  inside 
edge  measures  8  inches.    How  many  cubic  inches  are  in  the  chain  ? 

12.  In  a  stick  of  timber  50  feet  long,  and  7  x  10  inches,  there  are 
how  many  feet,  timber  measure  ?    How  many  feet,  board  measure  ? 

13.  Find  the  contents,  in  timber  measure  and  in  board  measure,  of  a 
stick  of  timber  18  ft.  long,  12  in.  wide,  15  in.  thick  at  one  end,  and  10 
in.  thick  at  the  other. 

14.  Find  the  side  of  the  largest  square  stick  of  timber  that  can  bo 
cut  from  a  log  2  feet  in  diameter.  17  indies^  nearly. 

15.  How  much  will  the  flooring  for  a  two-story  house  24  x  32  feet 
cost,  at  $40  per  M.,  the  flooring  being  1|  inches  thick  ?  $76.80. 

16.  How  much  lumber  in  a  stock  of  9  boards  13  ft.  long,  9  in.  wide, 
and  1^  in.  thick  ?  ^^^ft- 

17.  I  wish  to  have  built  in  my  cellar,  a  cistern  that  shall  hold  20 
lihd.,  and  to  have  it  a  cylinder  5  ft.  10  in.  in  diameter.  How  deep  ^vill 
it  be?  6  ft.  3.63  + in. 

18.  How  much  2-inch  plank  must  I  bay  for  a  5-foot  walk  on  the 
street  sides  of  a  corner  lot  4x8  rods,  the  walk  to  be  placed  2  ft.  6  in. 
from  the  fence  ?    And  how  much  will  it  cost  me,  at  $16  per  M.  ? 

19.  In  a  granary  is  a  bin  12|  ft.  long,  8  ft.  7  in.  wide,  and  5.4  ft. 
deep.    How  many  bushels  of  grain  will  it  hold  ? 

20.  What  is  the  capacity,  in  hogsheads,  of  a  rectangular  cistern  12  ft. 
long,  8  ft.  wide,  and  6  ft.  4  in.  deep  ? 

21.  A  laborer  built  a  wall  5  rd.  long,  5  ft.  thick  at  the  bottom,  2 
ft.  thick  at  the  top,  and  5  ft.  high,  in  2  days,  building  2  ft.  in  height 
the  first  day.    On  which  day  did  he  lay  the  most  wall  ? 

22.  If  a  trqugli  5  feet  long  holds  12  pailfuls  of  water,  how  many 
pailf uls  will  a  similar  box  hold  that  is  8  feet  long. 

23.  If  a  pint  of  wine  will  fill  15  cone-shaped  wine-glasses,  how 
many  times  will  a  gallon  of  wine  fill  a  similar  glass  of  1 }  times  the 
diameter  at  the  top  ?  35'1, 


CHAPTER  12. 

MISCELLANEOUS  PROBLEMS. 


S^^Tff^ 


1.  The  quotient  is  436fH,  and  the  divisor  is  735.  What  is  the 
dividend  ? 

2.  A  cubic  foot  of  water  weighs  G2  lb.  8  oz.  What  is  the  pressure 
on  1  acre  at  the  bottom  of  the  sea,  where  the  water  is  1,000  fathoms 
deep? 

3.  Find  the  prime  factors  of  .5313. 

4.  The  expenses  of  an  excursion  party,  consisting  of  8  gentlemen 
and  9  ladies,  were  $3.40  a  piece;  which  were  paid  by  the  gentlemen. 
How  much  did  each  pay  ? 

5.  $.725  is  what  fraction  of  $1  ? 

G.  How  many  feet  in  .735  of  a  mile  ? 

7.  If  the  cost  of  manufacturing  kip  boots  is  $4.60  a  pair,  and  they  are 
sold  at  25^  profit,  what  is  the  selling  price  ? 

8.  If  33 i-  lb.  of  tea  cost  $29^-,  how  much  will  12^^  lb.  cost  ? 

9.  How  much  lumber  in  a  stock  of  12  boards  14  ft.  long  and  10  in. 
wide? 

10.  A  grocer  bought  7  doz,  brooms  ©  $2.25,  and  retailed  them  at 
$".31^  apiece.     How  much  did  he  gain  on  the  lot  ? 

11.  One  week,  2,230  barrels  of  flour,  which  cost  $9.25  per  barrel, 
were  received  at  the  port  of  Cleveland,  and  it  was  sold  at  the  rate  of 
$3.15  per  sack  of  49  pounds.     What  was  the  gain  ? 

12.  How  many  pickets  each  3  inches  wide,  placed  3  inches  apart,  will 
it  take  for  a  fence  round  a  lot  4  x  10  rods  ? 

13.  What  is  the  shortest  distance  that  is  an  exact  number  of  times 
a  1-ft.  rule,  a  2-ft.  rule,  a  yard-stick,  and  a  10-ft.  pole? 

14.  At  $3.25  per  C,  how  many  broom  handles  can  I  buy  for  $26.52? 

15.  After  4,^  of  a  flock  of  sheep  had  been  killed  by  dogs,  and  68 
had  been  sold  to  a  butcher,  y  of  the  original  flock  were  left.  How 
many  sheep  were  in  the  flock  at  first  ? 

16.  If  21|  bushels  of  oats  are  required  to  seed  9|  acres,  how  many 
bushels  will  be  required  to  seed  a  field  of  17.2  acres  ? 

17.  IIow  many  sheets  of  tin,  each  14  x  22  in.,  will  it  take  to  cover  a 
roof,  each  side  of  which  is  80  ft.  long  and  18  ft,  4  in.  wide  ? 


330  MISCELLANEOUS    PROBLEMS. 

18.  A  printer's  price  for  "business  cards  is  $3.75  for  tlie  first  liundred, 
and  $1.25  per  liundred  for  any  number  after  the  first  liundred.  How 
much  will  1,500  cards  cost? 

19.  A  man,  dying,  leaves  an  estate  of  $53,1G6,  but  it  is  incumbered 
to  the  amount  of  $17,496.  His  widow  receives  ^  of  what  remains  after 
paying  the  incumbrances,  and  the  balance  is  divided  equally  among  5 
children.  What  is  the  widow's  share  ?  How  much  does  each  child 
receive  ? 

20.  A  fruit  dealer  pays  $4.25  per  bushel  for  3  bushels  of  chestnuts, 
and  sells  them  at  $.10  a  pint,  tin  measure.     What  are  his  profits  ? 

21.  What  fraction  equals  .00096  ? 

22.  A  grain  buyer  paid  $1.40  per  bushel  for  2,380  bushels  of  wheat, 
and  $.50  per  bushel  for  transportation  to  New  York,  where  he  sold  it 
at  a  loss  of  20;^.     How  much  did  he  lose  ? 

23.  If  25  cows  average  9  quarts  of  milk  each,  per  day,  through  the 
year,  and  it  is  sold  at  an  average  of  7  cents  per  quart,  and  the  expenses 
of  keeping  and  labor  are  $78  per  head,  what  are  the  annual  profits  ? 

24.  What  will  be  the  face  of  a  sight  draft  on  New  York,  that  costs 
$664  in  Louisville,  Ky.,  exchange  on  New  York  being  at  3|,^  premium  ? 

25.  I  invested  one  half  of  my  capital  in  bank-stock,  and  the  balance 
in  R.R.  stock.  I  gained  11^  on  the  bank-stock,  and  lost  7i%  on  the 
R.R.  stock,  and  my  net  gain  was  $175.    How  much  was  my  capital  ? 

26.  If  I  buy  1,600  bushels  of  oats  in  Iowa,  at  a  net  cost  of  $.56  per 
bushel,  transportation  included,  and  sell  them  in  New  York  @  $.54, 
how  much  do  I  gain  ? 

27.  A  grocer  paid  $22.75  for  a  barrel  of  mess  pork,  and  retailed  |  of 
it  at  $.12^  per  lb.,  and  the  balance  @  $.14.     What  were  his  profits? 

28.  Portland,  Me.,  is  in  latitude  43°  39'  north,  and  L.  Titicaca,  on  the 
same  meridian,  is  in  latitude  16°  42'  south.  How  many  miles,  air 
line  distance,  from  Portland  to  L.  Titicaca  ? 

29.  A  note  at  8  mo.  for  $750,  with  interest,  was  discounted  by  a 
Boston  bank,  3  months  after  date.    What  were  the  proceeds  ? 

30.  If  three  men  can  build  a  sidewalk  240  ft.  long  and  6  ft.  wide,  in 
15  days,  in  how  many  days  can  5  men  build  a  walk  180  ft.  long  and  4 
ft.  wide  ? 

31.  A  mechanic  contracted  to  work  a  year  for  $40  per  month,  his 
wages  payable  at  the  end  of  each  month.  Nothing  was  paid  him  till 
the  close  of  the  year,  when  he  received  the  whole  amount,  with  8,^ 
intorest.    How  much  did  he  receive  ? 


MISCELLANEOUS    PROBLEMS.  331 

32.  A  tax  of  $3,156  is  levied  on  a  Union  school -district,  whose 
assessed  valuation  is  $493,125.     What  is  the  rate  ? 

33.  In  the  above-mentioned  district  A's  property  is  assessed  at  $750, 
B's  at  $3,850,  C's  at  $1,600,  and  D's  at  $14,500.  How  much  is  each  one 
of  them  taxed  ? 

34.  Memorandum  : — Nov.  19,  1836,  gave  a  note  for  $1,650,  with  6^ 
interest.  June  18, 1868,  paid  $125  ;  Oct.  25,  1868,  paid  $475.  March 
4,  1869,  took  up  the  note.    How  much  was  the  last  payment  ? 

35.  How  many  squares  of  flooring  in  a  floor  44  x  75  ft.  ? 

36.  In  the  right  wing  of  an  army  there  were  18,675  men,  in  the 
center  23,518,  and  in  the  left  wing  11,498.  The  left  wing  was  re-in- 
forced  by  16,488  new  troops,  and  the  center  by  3,486.  The  command- 
ing general  then  ordered  9,894  men  from  the  left  wing  to  the  right, 
and  5,145  from  the  right  wing  to  the  center.  How  many  troops  were 
then  in  each  division  of  the  army  ? 

37.  A  pile  of  4-foot  wood  244  feet  long  and  5  feet  high,  was  sold  for 
$152.50.     What  was  the  price  per  cord  ? 

38.  What  is  the  commercial  weight  of  a  nugget  of  gold  that  weighs 
3  oz.  3  pvrt.  19i  gr.  ? 

39.  A  merchant  sold  broadcloth  at  5^  less  than  the  marked  price, 
and  yet  made  a  profit  of  25,^.  At  what  %  advance  on  cost  were  the 
goods  marked  ? 

40.  A  miller  pays  $1.45  per  bu.  for  225  bu.  of  wheat.  If  4.5  bu.  make 
1  bar.  of  flour,  and  he  sells  the  ship-stufis  for  $54.75,  at  what  price 
per  bar.  must  he  sell  the  flour,  to  realize  a  net  profit  of  $104.50  by  the 
transaction  ? 

41.  A  lOO-acre  farm  is  a  trapezoid  in  shape,  the  shorter  of  the  two 
parallel  sides  being  64.7  rods,  and  the  longer  135.3  rods.  How  far 
apart  are  these  two  sides  ? 

42.  How  many  panes  in  each  of  three  boxes  of  glass,  marked,  respec- 
tively, 8  X 10,  9  X 16,  and  10  x  18  ? 

43.  For  what  amount  must  I  draw  my  note,  payable  in  60  days,  to 
obtain  a  discount  of  $250  from  a  Philadelphia  bank  ? 

44.  A  teacher  receives  a  salary  of  $1,050,  and  6^  of  liis  expenses 
equals  20^^  of  his  savings.    How  much  does  he  save  yearly  ? 

45.  If  the  transportation  of  51.2  tons  of  freight  costs  $268.80,  how 
much  should  be  paid  for  the  transportation  of  32  J  tons  ? 

46.  I  paid  l{fc  for  an  insurance  of  $1,075  on  a  building  worth  $l,500l 
If  the  building  should  burn,  what  would  be  my  loss  ? 


332  MISCELLANEOUS    PKOBLEMS. 

47.  A  field,  wliicli  is  3^  times  as  long  as  it  is  wide,  contains  22.4 
acres.     What  arc  its  dimensions  ? 

48.  A  block  of  marble  contains  54H  cubic  feet,  and  tlie  leno-tli, 
breadth,  and  thickness  are  as  7,  4,  and  1.     What  are  the  dimensions  ? 

49.  I  mi.  +  71  rd.  +  ^  yd. = what  distance  ? 

50.  A  note  for  $35G,  dated  Mar.  10,  at  10  mo,,  with  interest  at  7^-, 
was  discounted  at  the  American  Exchange  Bank  of  New  York  City, 
Aug.  35.     What  were  the  proceeds  ? 

51.  A  drover  bought  135  head  of  cattle  @  $23,  and  147  head  @  $19, 
and  shipped  them  to  New  York,  at  a  cost  of  $1,597.  He  sold  163  head 
@  $37,  and  the  balance  @  $31.  Did  he  gain  or  lose,  on  the  whole  drove, 
and  how  much  ? 

52.  The  latitude  of  Chicago  is  41°  54'  N.,  and  Mobile  is  706A¥o 
miles  S.  of  Chicago.     In  what  latitude  is  Mobile  ? 

53.  I  sold  a  horse  for  ~g  more  than  he  cost  me,  receiving  $216  for  him. 
How  much  did  he  cost  me  ? 

54.  A  cannon-ball  15  inches  in  diameter  weighs  456  pounds.  What 
is  the  diameter  of  a  260-pound  shot  ? 

55.  A  mechanic  having  $852.75,  paid  f-  of  his  money  for  a  half-acre 
lot  of  land.     How  much  would  an  acre  cost,  at  the  same  rate  ? 

56.  A  sixty-day  note  for  $237.40,  dated  Poughkeepsie,  N.  Y.,  June  21, 
was  discounted  at  the  Second  National  Bank  of  Troy,  June  28.  How 
much  money  was  received  ? 

57.  How  far  is  it  from  one  of  the  lower  corners  to  the  diagonal  upper 
corner  of  a  room  20  ft.  long,  16  ft.  wide,  and  12  ft.  high  ? 

58.  When  it  is  9  o'clock,  A.  M.,  at  Cincinnati,  84°  24'  west,  it  is  9 
o'clock  47  min.  12  sec,  A.M.,  at  Montpelier.  What  is  the  longitude 
of  Montpelier? 

59.  A  farmer  exchanged  2  bu.  of  beans  @  $1.31^,  for  sugars  at  $.10 
and  $.11  per  lb.,  taking  the  same  quantity  of  each  kind.  How  many 
pounds  of  sugar  did  he  receive  ? 

60.  Wliat  is  the  interest,  in  this  State,  on  a  mortgage  for  $490,  for  1 
yr.  5  mo.  24  da.  ? 

61.  A  andB  were  partners  in  business,  with  a  capital  of  $1,250,  of 
which  A  furnished  $750.  At  the  end  of  the  first  year  A's  share  of  the 
profits  was  $340.65,  when  B  sold  his  interest  to  C  for  $637.50.  At  the 
end  of  the  second  year  C's  share  of  the  profits  was  $247.80,  when  he 
bought  A's  interest  for  $870.74.  How  much  did  A  and  B  each  make  ? 
And  how  much  had  C  invested  more  than  he  had  realized  ? 


MISCELLANEOUS    PROBLEMS.  333 

62.  From  the  product  of  the  sum  and  difference  of  3.G  and  3.24,  sub- 
tract the  difference  between  the  squares  of  3.6  and  2.24. 

63.  A  can  build  50  rods  of  fence  in  14  days,  B  can  build  it  in  25  days, 
C  in  8  days,  and  D  in  20  days.  In  what  time  can  they  build  it,  if  they 
all  work  together  ? 

64.  The  product  is  I,  and  the  multiplier  is  J  of  f  of  *^.  What  is  the 
multiplicand  ? 

65.  One  day  a  boy  bought  peaches  at  the  rate  of  3  for  4  cents,  and 
sold  them  at  the  rate  of  2  for  5  cents,  and  cleared  $4.20.  How  many 
peaches  did  he  buy  and  sell  ? 

QQ.  If  each  one  of  us  breathes  30  cubic  feet  of  air  per  hour,  in  how 
long  a  time  will  we  breathe  as  much  air  as  this  school-room  contains? 

67.  What  is  the  equated  time  for  the  payment  of  $100  due  in  6  mo., 
$120  due  in  7  mo.,  and  $160  due  in  10  mo.  ? 

68.  Pekin  is  in  118°  E.  longitiide,  and  San  Francisco  is  in  122°  W. 
longitude.  When  it  is  noon  at  Pekin,  what  is  the  hour  at  San  Fran 
Cisco  ? 

69.  The  floor  of  a  public  hall  56  x  84  feet,  is  of  boards  14  feet  long  and 
6  inches  wide,  which  are  nailed  with  10-penny  nails,  8  to  each  board. 
Allowing  68  nails  to  a  pound,  how  many  pounds  of  nails  are  in  the 
floor? 

70.  There  are  22 1^  bricks  to  a  cubic  foot  of  brick  wall.  What  part 
of  the  wall  consists  of  mortar  ? 

71.  How  many  bricks  in  the  walls  of  a  house  48  ft.  long,  25  feet  wide, 
and  18  ft.  high,  the  walls  being  1  foot  thick,  and  allowing  2^^  for 
openings  ? 

72.  If  2-^  yd.  of  cassimere  @  $1|  are  worth  as  much  as  .7  of  a  ton 
of  coal,  how  much  is  the  coal  worth  per  ton  ? 

73.  When  8  eggs  sell  for  $.25,  what  are  they  worth  per  dozen  ? 

74.  What  is  the  least  common  multiple  of  the  fractions  |,  {-,  f, 
and  I  ? 

Note. — Eeduce  the  fractions  to  least  similar  fractions. 

75.  What  is  the  least  common  multiple  of  222,  104,  68,  54,  and  34  ? 
What  is  the  greatest  common  divisor  of  the  same  numbers  ? 

76.  The  Oswego  Starch  Co.  drew  on  a  customer  in  Milwaukee  for 
$1,275,  at  60  days  after  sight.  The  bank  charged  ^%  for  collecting,  and 
required  2  days  for  transmission  each  way.  Exchange  on  Milwaukee 
being  at  !{%  discount,  what  were  the  proceeds  of  the  draft  ? 


334  MISCELLANEOUS    PROBLEMS. 

77.  At  $.36  per  sq.  yd.  for  plastering,  and  $.75  per  roll  for  paper- 
hanging,  liow  much  will  it  cost  to  plaster  the  walls  and  ceiling,  and 
paper  the  walls  of  a  room  18  x  16  x  9  ft.,  making  allowance,  in  paper- 
ing, for  3  windows,  each  3x6  ft.,  and  3  doors,  each  3x7  ft.,  the  wall- 
paper being  1  ft.  6  in.  wide  and  7  yd.  in  a  roll  ? 

78.  A  4-rod  road  extends  along  one  end  and  one  side  of  a  farm  which 
is  90.5  x  120  rd.,  the  farm  extending  to  the  middle  of  the  road.  How 
much  of  the  farm  is  in  the  road  ? 

79.  How  many  days  w^ill  it  take  a  ship  to  sail  from  St.  John's,  New- 
foundland, to  Valentia  Bay,  Ireland,  a  distance  of  1,950  miles,  if  she 
sails  at  the  rate  of  9.5  knots  per  hour? 

80.  What  length  of  a  board  9  inches  wide  will  make  a  square  foot  ? 

81.  At  $13.50  per  M,,  what  is  the  value  of  a  stock  of  13  boards  each 

14  ft.  long,  16  in.  wide  at  one  end,  and  tapering  to  a  point  ? 

82.  My  agent  in  Toledo  bought  5,000  barrels  of  apples  @  $1.60,  com- 
mission 2j%.    I  sent  him  a  draft  for  the  amount,  which  I  purchased 

^at  \^/c  discount.     I  paid  $.30  a  barrel  to  transport  the  apples  to  New 
York,  and  sold  them  @  $2.10.     What  were  my  profits  ? 

83.  In  a  straight  line  between  two 
buildings  standing  on  opposite  sides 
of  a  public  square,  is  a  post.  The 
building,  A,  is  55  ft.  high,  and  B 
64  ft.  From  the  foot  of  the  post  to 
the  base  of  the  building,  B,  is  76  ft.  ; 
from  the  top  of  the  post  to  the  top  of 

the  same  building  is  95  ft. ;  and  from  the  top  of  the  post  to  the  top  of 
the  building,  A,  is  80  feet.     What  is  the  height  of  the  post  ? 

84.  What  is  the  horizontal  distance  between  the  buildings? 

85.  What  is  the  distance  from  the  top  of  one  building  to  the  top  of 
the  other  ? 

86.  The  average  diameter  of  the  earth  is  9,111  miles.  How  many 
square  miles  on  its  surface  ? 

87.  Find  the  number  of  cubic  miles  in  the  earth. 

88.  If  6  masons  build  a  pier  35  ft.  long,  18  ft.  high,  and  4  ft.  wide,  in 

15  days  of  8  hours  each,  how  many  masons  will  be  required  to  build  a 
pier  48  ft.  long,  21  ft.  high,  and  5  ft.  wide,  in  12  days  of  10  hours  each  ? 

89.  Divide  an  estate  of  $7,500  among  3  children,  10, 12,  and  15  years 
old,  so  that  their  respective  shares,  at  Ifo  interest,  shall  amount  to 
the  same  sum  when  they  are  21  years  old. 


MISCELLANEOUS    PROBLEMS. 


335 


90.  A  mortgage  for  $13,275,  dated  St.  Louis,  Mo.,  Oct.  10,  1865, 
bears  the  following  indorsements  :  May  7,  1866,  $1,350 ;  Dec.  11,  18G6, 
$760;  June  23,  1867,  $500;  Nov.  8,  1867,  $850;  July  20,  1868,  $350. 
The  mortgage  was  taken  up  Jan.  1, 1869.  What  amount  was  then 
paid? 

91.  What  is  the  present  value  of  a  paid-up  lease  having  4  years  to 
run,  if  the  property  will  rent  for  $2,000  per  annum,  money  being 
worth  G%  compound  interest? 

92.  A  carpenter,  a  mason,  and  a  painter  built  a  house,  by  contract, 
for  $3,000.  The  carpenter  worked  108  days,  the  mason  72  -  days,  and 
the  painter  45  days,  and  the  materials  used  cost  $1,775.  How  much 
did  each  man  receive  for  his  labor? 

93.  Last  year  my  expenses,  which  were  80^  of  my  last  year's  income, 
equalled  9G%  of  my  expenses  this  year,  and  my  income  equalled  75^ 
of  this  year's  income.  Last  year  I  saved  $480.  How  much  do  I  save 
this  year  ? 

94.  A  broker  bought  115  shares  of  Express  stock  at  79 1.  He  ex- 
changed 63  shares  at  85  for  U.  S.  5-20's  at  111,  and  the  balance  at  i)ar 
for  RR.  stock  at  78.  He  afterward  sold  the  5-20's  at  116  L  and  the 
R.R.  stock  at  72.    Did  he  gain  or  lose,  and  how  much  ? 

95.  Find  the  balance  of  the  following  account,  and  the  equated  time 
for  its  payment : 

Dr.  Geo.  H.  Thomas.  (7r. 


1869 

1869 

Jan. 

13 

ToMdse.@4mo. 

23 

30 

Feb. 

35 

By  Cash, 

25 

00 

Feb. 

12 

«         ((            A    '( 

42 

83 

Apr. 

7 

"      " 

75 

00 

Mar. 

33 

"      «        6  " 

169 

33 

May 

33 

u 

30 

00 

Apr. 

19 

"      "  Cash, 

73 

19 

July 

7 

"  Note, 

75 

00 

June 

6 

"      «@30da. 

48 

53 

(( 

39 

"   Cash, 

35 

00 

Aug. 

16 

u           " 

50 

00 

96.  A  widow  who  is  left  with  a  daughter  16,  and  a  son  8  years  old, 
is  to  have  the  income  of  property  that  pays  an  annual  rent  of  $1,500 
above  taxes  and  repairs,  till  the  daughter  is  31  years  old.  The  daugh- 
ter is  then  to  have  the  income,  till  the  son  attains  his  majority,  when 
the  property  is  to  be  his.  How  much  is  each  one's  interest  in  the 
property  worth  to-day,  money  being  worth  Q%  simple  interest  ? 


PACK 

Addition  of  Compoiind  Numbers...  146 

"         "  Decimals 81 

"  Fractions 191 

"         "   Intesers 17 

Anal5'sis ." 223 

Aritlimetical  Progression 811 

Average  of  Payments 275 

anking 2G9 


Changes  of  Dividend  and  Divisor.. .  165 

Commission 241 

Common  Divisors 172 

'•        Multiples 176 

Compound  N  umbePvS 113 

Compound  Proportion 292 

Converse  Opeuations 205 

Converse  lleductions 210 

Cube  Eoot,  Extraction  of. 306 


Decimals 

Definitions  in  Compound  Numbers. 

"  "  Decimals 

"  "  Evolution 

"  "  Factors  and  Multiples 

"  "  Integers 

"  "   Mensuration 

"  "  Percentage 

"  "  Progressions 

Discount 

Division  of  Compound  Numbers.. . . 

"        "   Decimals ^ 

*'        "   Fractions 

"        "   Integers 

Divisors,  Common 


EVOLTTTION 800 

Exchange 273 

Extraction  of  Cube  Hoot 806 

"   Square  "    301 

Factors  and  Multiples 164 

Feactions 179 

Geometrical  Progression 315 

Government  Securities 267 

Insurance 238 

Integees 9 

Interest 252 

"        by  Progressions 319 

Longitude  and  Time 224 

Manual  of  Methods  and  Sugges- 
tions   5 

Measurement  of  Eight- Angled  Sur-. 
faces  and  Solids. 


paok 

Mensuratiox -{21 

Mensuration  of  Capacities 827 

"  "  Lines 323 

"  "  Surfaces 8-24 

Miscellaneous  Problems 329 

Multiples,  Common 170 

Multiplication  of  Compound  Nos 154 

"  "   Decimals 85 

"  "   Fractions 195 

"  "  Integers 35 

Notation  of  Compound  Numbers. . .  115 

"         "  Decimals 74 

"         "  Evolution 300 

"         "  Fractions 179 

"         "   Integers 10 

"        "  Percentage 229 

Partnership 294 

Percentage 229 

Percentage,  the  Five  General  Cases 

of 231 

Prefiice 8 

Price,  Quantity,  and  Cost 216 

Prime  Numbers 171 

Profit  and  Loss 243 

Progression,  Arithmetical 311 

"  Geometrical 315 

Progressions 31 1 

Properties  of  Composite  Numbers..  169 

Proportion,  Compound 292 

"  Simple 289 

Eatio  and  Proportion 2S7 

lleductions.  Converse 210 

"  of  Compound  Numbers.  115 

"  "  Fractions 183 

Eeview    Problems    in    Compound 

Numbers..  161 
"  "    Converse  Operations  226 

"  "    Decimals 110 

"  "    Fractions 203 

"  "    Integers 69 

"  "    Percentage 283 

"  "    Proportion 299 

Simple  Proportion 289 

Square  Eoot,  Extraction  of. 801 

Stocks 245 

Subtraction  of  Compound  Numbers.  149 

"  "  Decimals 83 

"  "  Fracfions 191 

"  "  Integers 26 

.•,rTax«s  and  Duties 243 

^  .4Jntt&-d  States  Money 95 


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